The 27-year-old Ivory Coast international has put pen to paper on a deal which will keep him at Old Trafford until at least 2024, and is targeting Europa League glory this season – United face Roma in the first leg of their semi-final on Thursday – before mounting a challenge for the Premier League title.
Bailly told the club's official website: "For me, it's the dream to be here. Now it's my fifth season and I hope to be here a long time.
"We are in a good position. We have the Europa League and we are second in the league, and that is very good.
"We hope we can win this trophy [the Europa League] – it is very important for us and for the fans and for the season. We want to finish well in the Premier League as well, I'm sure we will.
"I hope I can be ready and I don't get any injuries and to be strong and finish the season well. I hope to win the Europa League this season.
"We will have more challenges next season – hopefully the Champions League – and, for me, my dream is to win the Premier League, that's my big dream."
Asked if United could challenge for the domestic prize next season, he added: "I believe, I believe. I believe in my team-mates.
"We are a good, young team. The team wants to improve and that's very important for the group for next season."
Bailly, who is currently behind Harry Maguire and Victor Lindelof in the pecking order, joined United in a £30million switch from Spanish side Villarreal in June 2016 and was part of the team which won the Europa League in 2017.
However, his time in England has been punctuated by injury, and he has managed only 15 appearances to date this season, his most recent absence the result of a period of self-isolation after he contracted Covid-19 while on international duty.
Speaking about his contract, he said: "I'm very happy. This decision is something I didn't have to think about.
"I love this club and I love playing for Manchester United. I'm happy and so are my family, everything is good.
"Now the moment with my injury has passed and I'm fit and feel good, and that's it. A new contract is like a new challenge and I'm ready for this."
Manager Ole Gunnar Solskjaer, who has been linked with a summer move for Real Madrid's Raphael Varane among others, welcomed the defender's decision to extend his stay.
He said: "I am pleased that Eric has signed this new contract. He is still learning and improving all of the time under the coaches here.
"Eric has improved his robustness since my time as manager and he will continue to play a significant part in the squad."