An intense derby at Anfield appeared poised for a replay, when Luis Suarez would have been available to play, until Kuyt eluded the attentions of Red Devils captain Patrice Evra to hammer home two minutes from time.
Daniel Agger had earlier headed the home side in front before Park Ji-sung drew United level in a game which was evenly contested throughout.
Antonio Valencia was a constant threat on the right for United and he came close to breaking the deadlock on 17 minutes when the winger's shot following a penetrating run came back off the post.
However, Liverpool responded by taking the lead four minutes later when Agger took advantage of goalkeeper David de Gea's inability to take command of his penalty area to head home Steven Gerrard's corner.
The early stages of the second half saw both teams have hopeful appeals for a penalty dismissed by referee Mark Halsey before Martin Skrtel came to Jose Reina's rescue after Danny Welbeck had flicked the ball past the Liverpool keeper.
Liverpool upped the pressure in the final 10 minutes and were rewarded with two minutes to go when Andy Carroll flicked on Reina's goal-kick and substitute Kuyt thumped his shot beyond De Gea.
The game should have been put to bed a minute later when Carroll's header came back off the frame of the goal and Kuyt prodded the ball wide, but it made no difference as United failed to mount one of their famous late rallies.