Both sides have qualified for the 2026/27 UEFA Champions League, but after United activated his £35M release clause at Villa Park, the Belgium international instantly accepted their offer.
United had tried and failed twice before to sign Tielemans in his earlier career and the 29-year-old has signed a deal at Old Trafford until 2031.
As part of his first major interview since joining Michael Carrick's side, Tielemans praised United's status in the game, and claimed Villa just could not match that.
"I was not expecting much this summer. I was just focused on my football, focused on the World Cup and trying to perform my best for my country.
"But you know every summer something can happen and obviously I was aware of the clause.
"I'm very thankful to Aston Villa. I owe them the world. It's a great club, we did amazing things and they have put me back to my best level. But there are clubs above Villa, and Manchester United is one of them.
"It's a great achievement for any footballer to end up here. It means you have done well in your career. But it does not end there. I want to be successful."