Luis Diaz has scored six Premier League goals this season
Luton led at half-time through Chiedozie Ogbene's header as Liverpool, and Luis Diaz in particular, spurned chance after chance.
The Reds were rampant after the break and their pressure paid off when captain Virgil van Dijk showed desire and determination to get his side on level terms by powering home a header from a 56th-minute corner.
Just 125 seconds later Liverpool led as Cody Gakpo nodded in from close range as Alexis Mac Allister fired in a cross from the byeline.
Diaz, with his ninth shot of the game, finally got his reward for relentless work with the third goal in the 71st minute.
The Colombian forward danced his way into the box and coolly side-footed beyond Thomas Kaminski at the near post to give the hosts some breathing space on what had initially looked like a night when they could slip up in the title race.
Harvey Elliott, making his 100th appearance for the club, rounded off a fine second half when he pounced on a loose ball in the 90th minute to claim the fourth.
It made it three wins on the bounce for Liverpool after losing 3-1 at Arsenal last month and built further momentum as they gear up for Sunday's EFL Cup final against Chelsea at Wembley.
Jurgen Klopp's men boast a four-point cushion over second-placed Manchester City, although they have played a game more than the defending champions, while Luton remain third from bottom.
-- Courtesy of BBC Sport