The Dutchman is now Liverpool's third-highest scorer in this competition and they were more than thankful for his eventual winner in the final seconds of the first half.
Liverpool kept faith with the side that had produced their best performance on Saturday in beating Burnley 4-0, with Fabio Aurelio on the bench for his first involvement of the season following a knee injury.
But they were sluggish and uninspired and it took them an awfully long time in the first period to get some reward for their total domination.
In fact there were barely a couple of seconds left in the half to restart the match when Kuyt forced the ball home after Vukasin Poleksic had palmed out a Fernando Torres drive.
Liverpool improved after the break, which was not difficult, and Steven Gerrard smashed a swerving 25-yarder against the crossbar before Yossi Benayoun went on a brilliant run past a string of defenders before forcing Poleksic into a fumbling save at the foot of the near post.