Two more goals from Edinson Cavani confirmed Manchester United’s passage into the Europa League final - though Roma worked hard for their win in the second leg, contributing richly to a hugely entertaining tie that yielded 13 goals over 180 minutes.
United now meet Villarreal - who beat Arsenal - in Gdansk, and will fancy themselves to win their first trophy under Ole Gunnar Solskjaer and their first under anyone in five seasons. The sides have met four times before, achieving four goalless draws.
Though the tie was effectively settled in the first leg won by United 6-2, Roma started the second leg really well and forced David de Gea into a succession of smart saves.
Opinion: Could Rashford and Sancho stop people hating Man Utd?
But the suspicion remained that United had too much class, and sure enough they took the lead on the night when, following a couple of decent chances missed, Cavani finished emphatically following good work from Paul Pogba, Bruno Fernandes and Fred.
At the start of the second half, a game and dangerous Roma created a succession of chances, De Gea making several witting and unwitting saves.
Eventually, though, Edin Dzeko and Bryan Cristante beat him and for a few manic minutes things looked tight. But then another brilliant ball from Fernandes incited another brilliant finish from Cavani, this time a header, and both sides acknowledged the end of the contest, dropping the pace. But they still continued to attack, and on 83 minutes Nicola Zalewski’s volley went in off Alex Telles, giving Roma the consolation of a home win.
Encouragement for Villarreal? United are a better side than Villarreal, but when Villarreal watch tape of United's performance tonight, they will find encouragement. If Ole Gunnar Solskjaer's team are at their best, they will win, but if they are not - and this is their first final as a team - they can be careless in front of goal and iffy at the back. Last season, they lost to Sevilla at the semi-final stage for precisely those reasons.
Though United have improved hugely since then, performances like tonight's are the reasons Solskjaer tends to pick Fred and Scott McTominay in midfield for the big games, which forces him to omit an attacker, usually Mason Greenwood. This might be acceptable until the end of this season, but in the future that cannot be - Greenwood's quality demands it.
For that reason, United's summer priority should be a specialist number 6 - Fred is sometimes good but not good enough, while McTominay is a box-to-box player. A passer with a engine - not an easy thing to find - would turn a really good team into a brilliant team.
MAN OF THE MATCH
Edinson Cavani (Manchester United): David de Gea made some really good saves, but Cavani scored two, wore a plaster on his cheek, and has unbelievable hair. What can you do?
Roma: Milante 6, Smalling 6, Ibanez 6, Peres 5, Karsdorp 6, Pedro 7, Mancini 6, Cristante 7, Pellegrini 7, Mkhitaryan 8, Dzeko 7. Subs: Darboe 6, Santon 6, Mayoral 6, Zalewski 7.
Manchester United: De Gea 8, Wan Bissaka 6, Maguire 6, Bailly 5, Shaw 6, Fred 6, Pogba 6, Fernandes 7, Van de Beek 5, Greenwood 7, Cavani 8. Subs: Williams 5, Telles 5, Matic 6, Rashford 6, Mata 6.
There were 13 goals in the tie, the joint-second highest tally across two legs of any major Uefa competition semi-final. The record is 16: Eintracht Frankfurt 12-4 Rangers in the 1959-60 European Cup.
34’ - Roma have been a threat this half, and Dzeko does really well to hold off Bailly while making an angle for a shot ... but Bailly does well to get back at him, getting a touch that deadens the shot only for the ball to reach Pellegrini, who hammers a riser that De Gea blocks away.
39’ - GOAL! Roma 0-1 Manchester United (Cavani) agg 2-7 As I was saying, this a lovely goal! Pogba wins it deep and finds Fernandes, who flips a lovely ball around the corner with the outside of his boot. This allows Fred to drive forward and ... and ... and he curves a lush pass through to Cavani, with the outside of his right foot! This is not a misprint. Cavani then takes one fine touch to get the ball out of his feet and into his stride, then absolutely dematerialises a rising finish past Mirante. It's close enough so that the keeper gets a touch, but he's never stopping it. There's a quick check for offside which interrupts Cavani mid-arrow, before the ref confirms the goal.
57’ - GOAL! Roma 1-1 Manchester United (Dzeko) agg 3-7 Mkhitaryan, who's been good over the tie, dashes to the line and snaps over a low cross that Pedro arrives on the end of, shinning as slice into the middle where Dzeko, who loves scoring against United, turns in from a few yards. Some poor sap then feel obliged to dash into the net to retireve the ball.
60’ - GOAL! Roma 2-1 Manchester United (Cristante) agg 3-7 Fred tries to flick his way out of trouble - he's the least Brazilian Brazilian but he's still Brazillian - and Pellegrini robs him because he's also still Fred, then Cristante smashes a brilliant first-time drive into the far side-netting. Now then.
61’ - OH MY DAYS! De Gea makes two brilliant reaction saves that I barely saw because I was describing the goal, but United are rocking!
63’ - WHAT IS GOING ON! Roma have been good in the tie and looked dangerous all the way through; now that United have stopped playing, they look liable to score at any time, and now Pedro crosses low, the ball clips De Gea's inside heel, goes through his legs, and Shaw humps clear!
68’ - GOAL! Roma 2-2 Manchester United (Cavani) agg 3-8 What a player and what a run of form! Fred darts a ball into the box and Greenwood turns on it, laying back to Fernandes just outside the box, left of it. He looks up and over on the far side, Cavani knows, dipping into the space and attacking behind the defenders, powering home a downward header when inevitably immaculate cross arrives. That, surely, clinches a belting tie.
83’ - GOAL! Roma 3-2 Manchester United (Zalewski) agg 5-8 Ah! On debut! Santon goes down the left and crosses to the back post, where Zalewski semi-shanks home a volley off Telles. It looked like someone in the middle was offside, but no one seems mithered.
'This is where I belong' - Man Utd confirm Sancho transfer
Messages of support by Rashford's mural washed away