Derby day delight boosts Liverpool
Liverpool, whose hopes of reaching the knockout phase of the Champions League disappeared in midweek, took a fortunate early lead when Javier Mascherano's speculative shot was deflected past Tim Howard by Everton defender Joseph Yobo.
Everton had two goals ruled out for offside and were then denied by a great double-save by Pepe Reina before Dirk Kuyt finished it 10 minutes from time, seizing on Howard's parry of an Alberto Riera shot after another Yobo blunder.
The win takes Liverpool to fifth place on 23 points and leaves Everton two places above the relegation zone after a dire run.
Earlier on Sunday Birmingham City beat Wolverhampton Wanderers 1-0.