Manchester United returned to the Premier League summit by routing bottom-place Portsmouth 5-0 on Saturday, while Manchester City lost 2-1 at relegation-threatened Hull.
Wayne Rooney and Dimitar Berbatov each scored for United, while three Portsmouth players scored own goals to add to the debt-ridden club's woes.
United will now hope Chelsea is beaten by third-place Arsenal on Sunday so the champions retain their one-point lead.
While Man City's hopes of breaking into the top four were dealt a blow at Hull, rejuvenated Liverpool maintained its push for a Champions League place by beating Everton 1-0 in the Merseyside derby.
Tottenham had a chance to regain fourth place from Liverpool by beating Aston Villa in the late match.
After a lacklustre start to the season, Liverpool has now collected 17 points from a possible 21 in the last seven matches.
"Derbies are all about passion, commitment and fight, and we've beaten a good Everton side here with 10 men, so we should be proud and enjoy this," Liverpool captain Steven Gerrard said. "Our attitude prevailed, we stuck together and then took our chance when it came."
Despite Everton defender Phil Neville clinging to him, Dirk Kuyt sent Gerrard's inswinging corner past goalkeeper Tim Howard from close range in the 55th minute. It was Kuyt's 50th Liverpool goal.
Liverpool was a man down from the 35th minute after Sotirios Kyrgiakos was sent off for a two-footed challenge on Everton midfielder Marouane Fellaini, who escaped punishment for stamping on the Greek defender in the same incident.
Everton winger Steven Pienaar was shown a second yellow card in the 90th for a light challenge on Gerrard, but the South African was fortunate not to be sent off in the first half for a foul on Javier Mascherano.