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, would they?"
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."
Hull assistant boss Brian Horton, manager at Maine Road from 1993 to 1995, saluted his side's continued improvement following draws with Wolves and Chelsea.
Horton said: "The first half was as well as we've played for a long, long time. The Chelsea result has given them confidence and we look a completely different side.
"We should have beaten Wolves and to come back and do what they did against two of the best sides in the country is a tremendous testament to the staff and the fans."
While Mancini licked his wounds, the outlook was more positive across Manchester as United routed Portsmouth 5-0 and manager Sir Alex Ferguson said: "We had a lot of pressure and possession of the ball but we were patient enough."
The result came with the aid of three own goals and Ferguson added: "We were a bit fortunate but nonetheless you have to take them."
Pompey were left marooned six points adrift at the foot of the table, but boss Avram Grant insisted: "Positive thinking is always better than negative thinking and as long as we have the chance we will fight.
"For me it's a challenge. We need to do our job even if it feels like things are going against us."
They were joined in the bottom three by West Ham, the victims of Brian Laws' first win as Burnley manager.
"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.