Ryan Giggs opened the scoring for the hosts from the penalty spot but when Ledley King equalised with 20 minutes remaining Spurs looked to be heading away from Old Trafford with a point.
However, Nani scored to give United a 2-1 lead on 81 minutes and Giggs made it 3-1, from the penalty spot again, on 86 minutes.
The defeat against United and Manchester City's draw at Arsenal on Saturday evening kept Spurs in fourth place in the Premier League table but they are now just a point ahead of City with a trip to Eastlands amongst their remaining three games
"We had an opportunity to come and get something, but we didn't play as well as we had done in the last two games," Redknapp said after the game.
"With ten minutes to go at one-all, you're looking to come away from here with a point.
"We gave two sloppy penalties away today from poor defending and for the other goal we let Nani run and didn't defend well enough. It cost us in the end.
"It's all to play for still and it's going to be tight, it's going to go down to the last game I would imagine.
"It's very close and anyone can put a run of games together and can still make it.
"We've got to keep going. We've got Bolton at home next week, which won't be easy, then we've got Man City and then Burnley, so we need to pick up some points from those games and hope we can make it."