Liverpool fans pay tribute to Sean Cox in Rome after he suffered head injuries before the first leg against Roma
Roma owner James Pallotta has been banned from attending any Uefa game for three months - and his club fined 19,000 euros (£17,025) - after their Champions League defeat by Liverpool.
Pallotta was sanctioned for calling the match officiating an "absolute joke".
The club were charged over insufficient organisation and a blocked stairway at the match in Rome on 2 May, and for fans setting off fireworks.
Liverpool lost the semi-final second leg 4-2, but won 7-6 on aggregate.
Roma's fine follows a 50,000 euros (£43,632) penalty for crowd trouble at the first leg on 24 April. Liverpool fan Sean Cox suffered serious head injuries in an attack outside Anfield before that game.
Two Italian men have been charged with violent disorder in relation to the incident.
Alisson played in both legs as Roma lost 7-6 on aggregate to Liverpool in the semi-finals of the Champions League last season.
Liverpool have agreed a fee of about £66.8m for Roma goalkeeper Alisson.
The Premier League club are now expected to hold talks with the Brazil international.
The 25-year-old has been at Roma for two years, and played 37 Serie A games last season, after starting his career at Brazilian side Internacional.
Liverpool's bid, which is 70m euros, would overtake the current record for a keeper - 53m euros paid by Juventus to Parma for Gianluigi Buffon in 2001.
The Premier League record for a keeper is the 40m euros Manchester City paid Benfica for Ederson in June 2017.
Liverpool boss Jurgen Klopp was forced to defend goalkeeper Loris Karius last week after another mistake - this time in a pre-season friendly against Tranmere.
The 25-year-old German spilled a free-kick that allowed the hosts to score in Liverpool's 3-2 win at Prenton Park.
Karius was at fault for two goals in Liverpool's 3-1 defeat by Real Madrid in the Champions League final, but it was later found that he had concussion.
Alisson played all five games, keeping three clean sheets, as Brazil were knocked out at the quarter-final stage of the World Cup by Belgium.
Salah stars as superb Liverpool take control against Roma
Mohamed Salah is Liverpool's second-highest scorer in a season - only four goals behind Ian Rush
Another Mohamed Salah masterclass helped Liverpool take control of their Champions League semi-final - but two late Roma away goals gave the Italian side a glimmer of hope for the second leg.
Liverpool, who last reached the final in 2007, repeatedly breached the visitors' naively high defensive line and scored five times in the opening 68 minutes at a raucous Anfield.
Salah, who has now scored 43 goals since his summer move from Roma, scored twice and assisted two other goals.
He curled his first into the top corner from the edge of the box - then clipped the ball over Alisson for a second.
Salah then ran down the right and squared for Sadio Mane to make it 3-0 - before doing the same to set up Roberto Firmino.
Liverpool then became only the second team to score five goals in a Champions League semi-final match when Firmino headed in James Milner's corner.
With his side leading 5-0, Reds boss Jurgen Klopp took off Salah - and Roma gave themselves a chance for next week's return leg at the Stadio Olimpico.
First Edin Dzeko lashed home from Radja Nainggolan's pass, then Diego Perotti put a penalty into the top right corner after Milner handled Nainggolan's shot.
And Roma need only look to the last round for inspiration - they lost the first leg in Barcelona 4-1 before winning 3-0 in Italy to progress on away goals.
A repeat of the latter scoreline this time would take them into the final against Bayern Munich or Real Madrid.
Prior to Roma's late goals, the one sour note for Liverpool was the sight of Alex Oxlade-Chamberlain leaving the field on a stretcher. Klopp said later the England midfielder "probably had a really bad injury".
Salah's greatest moment yet?
Mohamed Salah did not celebrate his goals - but Roberto Firmino did
When Salah moved to Anfield last summer for £34m after two good seasons for Roma, nobody could have predicted what would follow.
The Egypt winger has scored 43 goals in 48 games for Liverpool - an average of one every 87 minutes - and is the Premier League's top scorer.
He has netted in his past seven games - including four goals against Watford, and one in both legs of the Champions League quarter-final win over Manchester City. On Sunday, he was named Professional Footballers' Association Player of the Year.
But it has all been leading up to this night at Anfield.
Roma had no answer to their former player, who opened the scoring off the underside of the bar after Jordan Henderson won the ball from Dzeko, and Mane and Firmino combined to find Salah.
His second came when he found Firmino from the halfway line, then ran forward to latch onto his team-mate's pass before lifting the ball over Roma keeper Alisson, considered one of the best in the world. He did not celebrate either goal.
Salah then took advantage of Roma's high defending to get in down the right, perhaps from an offside position, before finding the previously wasteful Mane as Liverpool started to cruise.
He then rolled a ball across the six-yard box for Firmino to score.
With the tie seemingly won, Salah came off to a standing ovation when he was replaced by Danny Ings with 15 minutes left.
But he might need to score again in Rome after the visitors' late rally.
Barcelona were far from their best in beating Roma 4-1 at the Nou Camp
Barcelona were far from their best but managed to secure a 4-1 first-leg victory at the Nou Camp to take full control of their Champions League quarter-final tie with Roma.
Ernesto Valverde's side struggled for rhythm for much of the game against the Italian side, but built a three-goal advantage thanks to own goals either side of half-time from Daniele De Rossi (40) and Kostas Manolas (55) and a tap in from Gerard Pique (59).
Edin Dzeko gave Roma a late glimmer of hope heading into next week's second leg at the Stadio Olimpico by scoring an away goal, but Luis Suarez (87) landed a late blow as Barca put one foot in the last four.
Barcelona enjoyed a 6-1 win over Roma in their last Champions League meeting in November 2015 and they might have thought they were on their way to another comprehensive win when Suarez had the ball in the back of the net in only the seventh minute, but the strike was correctly ruled out for offside.
It was the tale of the two former Premier League strikers in the early stages as, minutes later, Dzeko tumbled in the Barca box under a challenge from Nelson Semedo, but Dutch referee Danny Makkelie waved away the appeals.
The chances continued to fall for the hosts as Ivan Rakitic's strike rebounded off the post before Alisson diverted Suarez's goalbound strike beyond the far post.
However, they eventually took the lead just before the break and the goal came in fortunate circumstances. De Rossi slid in ahead Lionel Messi - who would have been in on goal - with an excellent challenge just inside the penalty area but unfortunately for the Italian, the ball ended up beating Alisson and nestling in the bottom corner.
After the break, Roma missed a glorious chance to pull one back when Diego Perotti headed wide from close range and they were punished when Pique tapped home the rebound from close range after Alisson palmed away Suarez's shot.
Barcelona needed Marc-Andre ter Stegen to make superb saves to deny Gregoire Defrel and Perotti, but the visitors eventually got their away goal 10 minutes from time.
Medhi Benatia's strike keeps the pressure on Serie A leaders Napoli over Christmas
By BBC Sport
Juventus defender Medhi Benatia scored the winner against his former club Roma as his side narrowed the gap on Serie A leaders Napoli to a point.
Benatia converted the rebound at the second attempt after Roma goalkeeper Alisson saved Giorgio Chiellini's header from a corner.
Striker Gonzalo Higuain should have doubled the champions' lead but his close-range effort flew over the bar.
Roma, who are fourth, hit the bar through Alessandro Florenzi.
Roma have now lost their last seven top-flight trips to Turin, not winning there since January 2010.
Eusebio di Francesco's side have failed to take any of the nine points available against their top-four rivals so far this season.
They had their chances late on against Juve, with Edin Dzeko's flicked header going wide on 87 minutes.
Then Juventus' former Arsenal goalkeeper Wojciech Szczesny denied Czech Republic striker Patrik Schick with a fine save after he went clean through on goal in stoppage time.
Earlier on Saturday, Marek Hamsik scored his 116th goal to overtake Diego Maradona as Napoli's all-time record goalscorer, and keep the Serie A leaders top for Christmas, while third-placed Inter and 11th-placed AC Milan both lost.
Hamsik's strike before the break ensured a come-from-behind 3-2 win over Sampdoria to give Maurizio Sarri's side 45 points from 18 games.
