
The result moves the Reds up into third position on 19 points,ahead of Manchester City on goal difference, and three points behind leaders Arsenal, who play later today at home against Bolton. Everton remain in 10th place.
Everton had the better of the first half and their increasing pressure and aerial threat eventually brought the game's opening goal. In the 37th minute Liverpool defender Sami Hyypia sliced the ball into his own net after his team's defence had failed to clear the ball.
Liverpool equalized in the 53rd minute through Kuyt's first penalty after Tony Hibbert's professional foul brought down Steven Gerrard in the box. After Hibbert received the red card, Kuyt scored to goalkeeper Tim Howard's left.
The away side were able then to apply more pressure throughout the second half, but they wasted several chances as it appeared Everton would hang on for a point.
Liverpool manager Rafael Benitez's decision to substitute captain Gerrard appeared particularly puzzling but another late penalty from Kuyt saved his blushes.
Two minutes into injury time, Everton captain Phil Neville was sent off after he handled the ball on the goal line to prevent substitute Lucas Leiva from scoring, and Kuyt scored his second to clinch a thrilling victory.
Manchester Utd will look to keep pace with Arsenal at the top as they take on Aston Villa at Villa Park for the evening kick-off.