First-half goals from Sadio Mane and Georginio Wijnaldum and Mohamed Salah's 48th-minute effort gave Liverpool a 3-0 win over Bournemouth at Anfield.
Victory meant Liverpool, who had been second on goal difference, leapfrog Manchester City by three points ahead of the reigning champions' match at home to Chelsea on Sunday.
Meanwhile United's revival under caretaker manager Ole Gunnar Solskjaer continued with an equally comfortable 3-0 win away to Fulham.
Liverpool had dropped points in successive 1-1 draws against Leicester and West Ham, amid suggestions that Jurgen Klopp's men were losing their nerve in a quest to end the club's 29-year wait to be crowned champions of England once again.
Mane, however, eased the pressure with an early headed goal at Anfield.
Liverpool doubled their lead through Wijnaldum's delightful 34th-minute lob before Egypt striker Salah sealed victory with a curling shot.
"The crowd was immense," said Liverpool's James Milner. "It's been a tough few weeks with players injured and ill. Georginio Wijnaldum was nowhere to be seen yesterday and then he produces a performance like that."
In London, Paul Pogba scored two goals - including a second-half penalty - either side of an emphatic Anthony Martial strike, as United made it 10 wins in 11 games since former striker Solskjaer took over.
"The top four is a big thing. There are so many games left, but we hope to stay in the top four," said Solskjaer.
Arsenal inflicted more misery on Huddersfield with a 2-1 win away to the bottom-of-the-table club.
Alex Iwobi volleyed the Gunners into an early lead and Alexandre Lacazette made it 2-0 before half-time, with Sead Kolasinac's late own-goal no consolation for Huddersfield.
Elsewhere Everton manager Marco Silva's return to Watford ended in a 1-0 defeat.
Watford striker Troy Deeney had scorned Silva in the build-up by saying "the people at Everton are fantastic, not the manager" and Andre Gray's first goal since September meant Deeney did not have to eat his words.