Rafael Benitez hailed Saturday's 1-0 Merseyside derby victory as a "massive" result in Liverpool's season as Tottenham boss Harry Redknapp admitted the weekend belonged to his side's Champions League rivals.
Dirk Kuyt's header gave the Reds victory over Everton and they were further boosted when Spurs and Aston Villa shared the spoils in the evening game.
Benitez said: "It was massive for us. To win in a derby is really important, to win in a derby with 10 players at Anfield is more important.
"Everyone was working so hard that I think at the end we deserved to win."
Opposite number David Moyes said of a game that finished 10 a side: "I think there have been lots of derbies very similar: quite feisty and that's why fans like them.
"We certainly didn't deserve to lose the game. We might not have deserved to win it but we shouldn't have lost it."
While Aston Villa boss Martin O'Neill, whose side are also in the hunt for the top four, stressed the goalless draw at White Hart Lane "keeps us alive," Redknapp felt Benitez's men were the real winners.
He said: "Obviously it was a big result for Liverpool. It's their weekend this week, it will be somebody else's next week, somebody else's the week after.
"It's going to go all the way to the wire. You didn't see Manchester City getting beat at Hull, not many people would have tipped that one."
Indeed they would not, and manager Roberto Mancini admitted the Blues' failings after Jozy Altidore's first Premier League goal and veteran George Boateng's first for the Tigers saw his side beaten 2-1.
Mancini said: "We didn't play very well. We started the game very, very slowly. "If we want to make the top four we must win away and we must fight. We must be aggressive away, not like the first half."
Portsmouth were joined in the bottom three by West Ham, the victims of Brian Laws' first win as Burnley boss. "It was a huge three points and when you talk about games you have to win this was one of them," said Laws. "There was so much pressure but the players were magnificent."
Hammers boss Gianfranco Zola would not press the panic button, but said: "We have to be quick because time is running out and the table requires it. But I'm not worried -- I'm disappointed because we had enough to do better."
Elsewhere, Stoke claimed a much-needed 3-0 win to increase the pressure on Blackburn, Kenwyne Jones' equaliser earned Sunderland a point against Wigan and Bolton's Kevin Davies had a late header harshly disallowed in a goalless draw with Fulham.