The Greece international was sent off 11 minutes before the break after a clash with Marouane Fellaini which resulted in the Belgian being carried off.
But Liverpool refused to buckle and Kuyt struck the decisive blow in the 55th minute when he headed in a Steven Gerrard corner and Everton's misery was complete in stoppage time when Steven Pienaar was sent off for a second yellow card.
Jamie Carragher set the tone in the opening two minutes when he was lucky to escape with just a stern word from referee Martin Atkinson after a late tackle on Pienaar.
Reds centre-back Daniel Agger smashed the first chance over the bar in the 15th minute before Pienaar was first into the book for clattering into Javier Mascherano.
Carragher was then booked for a bad challenge on South Africa international Pienaar before the match finally boiled over ten minutes before the break.
Fellaini and Kyrgiakos flew into a challenge with the Greece international seeing red and Fellaini being carried off after lengthy treatment.
A minute before the break Gerrard smashed a 25-yard free-kick onto the bar before Tim Cahill wasted the chance of the half at the other end.
Landon Donovan's cross was met by Mascherano but his clearance only found Cahill. The Australia international, usually so reliable with his head, nodded over from six yards.
And he was made to pay for that miss soon after the restart. Gerrard's corner found Kuyt in the six-yard box and he had the simple task of heading beyond Tim Howard.
But despite having the extra-man advantage Everton struggled to create any clear-cut chances in the second period.
The best of those came two minutes from time when Yakubu turned on the edge of the box and arrowed a shot towards the top corner but Pepe Reina brilliantly tipped the ball over the bar.
Pienaar was then sent for an early bath deep into stoppage time after jumping into Gerrard to pick up his second yellow card as the Reds held out.