The England international striker completed his medical at Old Trafford to complete possibly one of the most controversial transfers of recent years.
As a gesture of goodwill, Smith agreed to waive his signing on fee, while no agent fees will be charged either.
That will act as the balance of the transfer, with Leeds receiving a straight £6m plus Reds youngster Danny Pugh.
Gerald Krasner said after the move: "We are beginning a major programme of rebuilding and restructuring at Leeds and we both believe his immediate future in football lies away from Elland Road.
"We know the feelings he has for our club and supporters and what it means to him to leave the no 17 shirt behind. We thank him for his service to Leeds and wish him well in his future career."
Smith admited it was hard to leave Elland Road, but added that Manchester United was now the club for him.
"I have grown up at Leeds and spent some wonderful years there and would like to thank everyone associated with the club," he said. "Now I feel it's the right time to move on and there could be no better club than Manchester United.
"To be joining one of the best clubs in the world is fantastic and I'm looking forward to the new challenges that face me."
Sir Alex Ferguson said: "I have never been more impressed with a young player, his desire to play for Manchester United is fantastic.
"Certain young people come along with a special determination, and after speaking to him I expect that from Alan Smith. That desire will take him a long way."