The 24-year-old England international midfielder cited his ambition to win trophies as the key motivation for his move north to Eastlands.
After moving in a deal that takes Stephen Ireland to Villa Park, he said: "It didn't take too much selling by the boss (Roberto Mancini) when you see the players who are already here and the plans for the club.
"The last thing he said to me was 'be ready for the next game', and then he said 'be ready to win the Premier League', and that's my ultimate aim - that's what I want to do and why I'm here.
"There's a lot of competition to get there, it's where we all want to be. I'm ambitious and I want to win trophies here. I believe this is a great place to do that and I believe we will do it. Hopefully, sooner rather than later."
Milner's transfer was first mooted before this summer's World Cup, where he featured in England's campaign.
After securing a five-year deal, he told the club's website: "The move seems to have gone on a long time - now all I want to do is get out on the field and play.
"I'm ready and raring to go now we've finally got everything done. I can't wait to start a new chapter."
Milner joins a new illustrious cast, following the path taken by David Silva, Mario Balotelli, Jerome Boateng, Yaya Toure and Aleksander Kolarov.
Asked if Milner was his final signing, manager Roberto Mancini said: "I think so. He is a good young player, who can play in every position in the middle and out wide."
Manchester United manager Sir Alex Ferguson complained about the "kamikaze" attitude of some clubs in the transfer market although he did not mention City by name.
Mancini said: "If you want to buy a new player, you must spend money. It is the market.
"I respect his (Ferguson's) opinion but Manchester United, not Manchester City, spent a lot of money in the past."
Milner's future had been the subject of ongoing speculation since Villa rejected a bid of around £20million from City in May.
The two clubs finally agreed a fee but the deal was held up still further by the involvement of Ireland, who reportedly demanded a £2million pay-off from City.
Milner replaces Ireland as City's number seven and Mancini said: "I hope things go well for Stevie at Villa.
"He is also a very good player and he has played a big part in City's history. I think he will enjoy the change of being at another club and I hope he does well for Aston Villa."
Milner will sit out City's Europa League play-off first leg match on Thursday night against Timisoara and instead could make his debut against Liverpool on Monday.