Steven Gerrard's hat-trick sank Luton and lifted the spirits of embattled Liverpool boss Rafael Benitez.
Liverpool hat-trick hero Steven Gerrard celebrates with Sami Hyypia
Ryan Babel's powerful finish broke the deadlock in first-half injury time, with Gerrard heading home from close range after 51 minutes.
Sami Hyypia turned in Gerrard's corner six minutes later before the captain slid in his second from 14 yards.
Gerrard's third came via a swerving shot to seal an FA Cup fourth round tie with Swansea or Havant & Waterlooville.
Both sides formed a guard of honour before kick-off as Liverpool defender Jamie Carragher celebrated his 500th appearance for the club.
The Kop also cheered Benitez after revelations that American owners Tom Hicks and George Gillett had held secret talks with Jurgen Klinsmann to line him up as a potential successor to the Spaniard.
Luton fans were in good heart after administrator Brendan Guilfoyle revealed a deal has been agreed with a consortium, fronted by television presenter Nick Owen, to buy the club.
Liverpool, fielding a strong side including Fernando Torres, exerted total domination in the first 45 minutes.
But Luton, marshalled by former Anfield veteran Don Hutchison, defended heroically, putting their bodies on the line for a succession of brave blocks.
Peter Crouch should have done better than direct a fifth-minute header wide, and Babel curled a fine right-foot effort against an upright three minutes later.
Babel also had a shot deflected wide before he broke Luton's brave resistance in first-half injury time.
Gerrard's flick released Torres, and he set up the unmarked Babel for a drilled finish into the bottom corner past Dean Brill.
And Liverpool were home and dry after 51 minutes as Luton's lack of height at the back was ruthlessly exposed.
Jermaine Pennant's cross found Crouch at the far post and he knocked the ball into the path of the onrushing Gerrard, who headed beyond Brill.
Liverpool added a third six minutes later when Hyypia headed in Gerrard's near post corner, via a deflection from Matthew Spring.
Gerrard added his second after 64 minutes, steering home a precise finish from 14 yards after Torres' shot had been partially cleared into his path.
And a swerving long-range drive deceived Brill to complete Gerrard's hat-trick after 71 minutes, leaving him to depart to a standing ovation from Anfield.
Liverpool manager Rafael Benitez on co-owners secret talks with Jurgen Klinsmann:
"We weren't really thinking about it because we know it's important to progress in the FA Cup.
"I was not thinking too much about this, I was thinking about the game. I'm really happy because we won 5-0 and progressed in the competition.
"It wasn't a bad week because we were training normally. We knew everyone was talking but we were trying to concentrate on the game against Luton. I know people must ask but I want to talk about the game."
Luton Town manager Kevin Blackwell:
"They were square pegs in round holes but they gave me everything they have got.
"There is no doubt they (Liverpool) have ability in abundance and have done exceptionally well for Rafa.
"It was an awesome team they put out. Clearly they have done a great job for the manager."
Liverpool: Itandje, Arbeloa, Carragher, Hyypia, Riise, Pennant, Alonso, Gerrard (Lucas 73), Babel, Crouch (Kuyt 65), Torres (Aurelio 73).
Subs Not Used: Martin, Kewell.
Goals: Babel 45, Gerrard 52, Hyypia 57, Gerrard 64, 72.
Luton: Brill, Jackson, Hutchison, Keane, Goodall, Bell (McVeigh 73), Robinson (O'Leary 68), Spring, Currie, Andrew (Furlong 68), Talbot.
Subs Not Used: Parkin, Emanuel.
Ref: Peter Walton (Northamptonshire).
BBC Sport Player Rater man of the match: Liverpool's Steven Gerrard 8.33 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.