He will join up with United on June 30 when his current contract with the Welsh club expires. Robinson has scored 31 goals in the last three seasons.
Robinson, 28, wanted to wait until after Leeds' play-off final against Doncaster on Sunday before deciding whether to stay with the Swans or leave.
But a Swansea statement said: "The club felt it was an unrealistic request and imposed a deadline of yesterday for Andy to make his mind up. This he did, subsequently deciding to move on."
Scouser Robinson, joined Swansea in the summer of 2003 on a free transfer after being released by Tranmere.
He came close to leaving two years ago when contract talks stalled and bids came in from Southend and Cardiff.
But he eventually agreed a new two-year deal which expires next month. The Swans confirmed the latest round of contract talks started last October and had hoped to agree terms before the January transfer opened but they failed to reach an agreement.
"At that point, it was decided not to place Andy on the transfer list as we felt he had a great part to play in our quest for promotion," the statement added.
"The prospect of achieving that promotion with Andy as part of the squad far outweighed any financial gain we may have realised should Andy have moved on at that point.
"It was also felt that having gained promotion, it would be enough to further secure Andy's services. Negotiations have been ongoing and the latest offer, had it been accepted, would have made Andy one of the highest paid players in the history of Swansea City FC.
"The plan this summer was to keep every player who played a part in our promotion success and to build on that by further strengthening the squad.
"To that end, Roberto Martinez is currently working tirelessly to bring in the type of player that will add to the strength and character of the squad, with the aim of making Swansea City a competitive Championship club.
"Andy leaves Swansea City with our heartfelt best wishes and with many thanks for his service to the club and the commitment shown by him over the years that he has been at the club," the statement concluded.
"I'm delighted to be joining Leeds," said Robinson. "It's a massive name in football and a massive club, and it's great that I'm going to be a part of that.
"I've had an unbelievable five years at Swansea. I can't thank the club enough for what they have done for me and the fans have been brilliant.
"This was the toughest decision of my life, but it's right for me. I spoke to Roberto Martinez on Tuesday and told him of my decision.
"I didn't want this dragging on because I have the utmost respect for Roberto and for Swansea City."