United's 2-0 win over Ajax in Stockholm came two days after 22 people were killed by a suicide bomber following a concert at Manchester Arena.
The club tweeted a photo of a banner reading: "Manchester - A City United."
United manager Jose Mourinho said: "If we could, we would change obviously the people's lives for this cup."
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Manchester-born striker Marcus Rashford said the win was "for you Manchester," while the world's most expensive player Paul Pogba said: "We played for the people who died."
Captain Wayne Rooney encouraged people to donate to help the families of the victims; Juan Mata said the players had "focused on the game and to win and to lift the spirit of the town at least for some hours - and we feel proud of that" and midfielder Ander Herrera said: "I want to dedicate this trophy to the victims."
He added: "What happened two days ago was horrible. Yesterday morning we were devastated. It was difficult to train but the manager told us the only thing we could do is win this for them, and that is what we have done.
"We are just football players but we have an audience so we want to work together for a normal world. This happened in Manchester but everywhere we want to see a united world and fight for peace."
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On an emotional night in Sweden, Pogba and Henrikh Mkhitaryan scored the goals that won the Europa League for the first time in United's history - and with it a place in the group stages of next season's Champions League.
Officials considered postponing Wednesday's game following the bombing 48 hours earlier, but it was decided it should go ahead. However, United did cancel their pre-game news conference.
United boss Jose Mourinho said: "It is really hard and that is why we didn't want to have a press conference because we had to prepare ourselves to do our job and try to forget events that are obviously much bigger than our job.
"But the world goes on. It doesn't stop and we have to do our work. The players put a wall up and stayed isolated from everything and did really well.
"I agreed with the decision to play the game, but if we could exchange these lives we would do it immediately. We wouldn't think twice."
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United's victory in Stockholm means they have now won all three of Europe's major competitions, having previously won the Champions League three times - in 1968, 1999 and 2008 - and the Cup Winners' Cup in 1991.
The Europa League is their second major trophy of Mourinho's debut season in charge, having also won the League Cup in February.
However, the Old Trafford club finished sixth in the Premier League and their Portuguese manager has often appeared frustrated.
The former Chelsea boss recently said he has had to work "harder than ever" this campaign and his delight was clear to see at full-time.
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He was hugged by his son and thrown into the air by his backroom staff before embracing his players on the pitch and applauding the United fans inside the Friends Arena.
His side produced a professional, counter-attacking display with just 31% possession and afterwards Mourinho said: "There are a lot of poets in football but poets don't win many titles."
When asked about this in his post-match news conference, he said: "The poets are the ones that win every match. I think I am going to try to have one of these jobs. I never managed in my career to win every match - I always lose matches.
"In a bad season - in a season where sometimes I felt that my team was the worst team in the world, where I felt that sometimes I was the worst manager in the world - we managed to win three trophies.
"We go to the Champions League by winning a trophy, not by finishing second, third or fourth, and we have the honour of going to the Super Cup.
"So I think the season was really good. A victory of pragmatism, of humble people who respect opponents and try to stop their qualities and exploit their weaknesses.
"Not poets, just humble people."
United top scorer Zlatan Ibrahimovic missed Wednesday's final as a result of the serious knee injury he suffered in the Europa league quarter-final against Anderlecht in April, which ended his season.
The 35-year-old former Sweden striker, who was born in Stockholm, is yet to agree an extension to his one-year deal with the club.
"It is so unfair what happened to him. Sometimes football is unfair. Sometimes life is unfair," added Mourinho.
"This would have been amazing for him to be on the pitch in his home town.
"I'm really, really sorry that he could not play in this final - but he played outside with the other players who were not playing. In Swedish he was telling the ball boys to take it easy in the second half.
"I am so pleased he gets this final in his amazing career and he finally gets a European trophy."
Pogba and Mkhitaryan scored in each half to earn United victory on Wednesday.
Pogba, who returned to Old Trafford from Juventus last summer for a world record fee of £89m, saw his shot from the edge of the box deflected in off Davinson Sanchez in the 18th minute.
Fellow midfielder Mkhitaryan flicked the ball in from close range for his sixth goal in the Europa League this season three minutes after the break.
"We did it. We are very proud. Nobody can talk. They said we had a bad season but we have three trophies [including the Community Shield]," France international Pogba, 24, told BT Sport.
Armenia captain Mkhitaryan added: "Thank you to my team-mates because without them I couldn't score so many goals.
"After the last game against Celta Vigo I was dreaming about scoring in the final - and I did."
Juan Mata, who missed seven weeks of the season through injury from late March to early May, said: "We have won a trophy the club has never won so we are really happy.
"It has been a really difficult week in the city of Manchester and we are happy to win for all the people that have suffered."
Manchester United dedicated Wednesday's Europa League victory to the victims of Monday's attack in the city, as the club made tributes in their memory.