Falcao started repaying some of the money United have invested in him by scoring the winner in Sunday's 2-1 victory over Everton at Old Trafford.
The Colombian, who also cleared a Phil Jagielka effort off the line in the second half, is on loan from Monaco until the end of the season.
United will have to stump up £43million if they want to sign him permanently next summer and the Colombian hopes they do.
"Me and my family are happy here, and we would like to stay for many years at Manchester United," Falcao told MUTV.
Falcao's strike lifted United into the top four for the first time this season.
Their manager Louis van Gaal told his players after the final whistle that they are still playing well below the standard he requires though.
Van Gaal endured a nervy final few minutes in the home dugout as David de Gea, who earlier saved Leighton Baines' penalty, pulled off two stunning saves to deny Leon Osman and Bryan Oviedo.
And the Dutchman thinks his team gave too many chances away, particularly in the second half, and will demand an instant improvement when the players return from international duty.
''We are not playing good but we are already fourth in the table,'' the United manager said
''That is what I have told my players.
''We played well for one half
We have to do it 90 minutes.
''We haven't defended well
We have to improve our organisation.
''We have given in the second half four big chances away."
Van Gaal's opposite number Roberto Martinez hit out at Kevin Friend after the match, accusing the referee of costing Everton a point thanks to an apparent refereeing blunder.
Martinez believes the winning goal should not have been allowed to stand though as he felt the game should have been stopped to allow Steven Pienaar to receive treatment for an injury he sustained moments before Falcao found the net.
''We're disappointed with the second goal
I thought the referee made a major, major error,'' the Everton manager said.
''Steven Pienaar was on the ground, we made it clear to the referee to stop the game and he allowed it to play on and it was very, very disappointing.
''Steven receives a knock and he goes to ground, the ball is in front of him and the referee is well aware of it
He is talking with the fourth official.
''It is a clear-cut decision
The law says if the player is on the ground and the ball is not in a dangerous position you need to stop the game.
''Man United attacks and the ball ends up in our net and I don't think that is fair
I think that action was a big mistake from the referee.''
Martinez said he expects centre-back John Stones to be out for "a few weeks" with suspected ankle ligament damage, which occurred in the dying minutes of Sunday's game.
The Football Association confirmed on Sunday night that the defender had pulled out of the England squad for this week's Euro 2016 qualifiers against San Marino and Estonia because of the injury.
Source : PA
Source: PA