Marcelo Bielsa's side had trailed 2-0 early in the game at Elland Road but came roaring back to take the three points, while rival West Brom tripped up 2-1 at Cardiff City.
Third-placed Nottingham Forest is two points adrift of the struggling Baggies after edging out Brentford.
Wayne Rooney's first goal for Derby County, helped by a big deflection, gave Phillip Cocu's side the lead at Luton Town, but the former England captain finished on the losing side as the Hatters came back to score a 3-2 victory.
Blackburn was 2-1 victor against QPR, Huddersfield Town won 2-1 at Hull City, Bristol City recorded an away-day success at Reading, and lowly Wigan saw Jamal Lowe grab a stoppage-time winner to record a 2-1 triumph over Sheffield Wednesday.
BAMFORD THE HERO AS LEEDS RISES AGAIN
It would have been reasonable to ask if the wheels were coming off the Leeds bandwagon when it fell 2-0 behind against Millwall inside 23 minutes, as Shaun Hutchinson and Jed Wallace gave the visitor a dream start.
There have been worrying signs for Bielsa's Leeds in recent weeks but just as thoughts turned to their previous failures to capitalise on strong starts to league campaigns, the West Yorkshire side roused itself.
Bamford began the recovery by striking from close range in the 48th minute, Pablo Hernandez cracked an equaliser just after the hour, and Bamford had Elland Road rocking when he stooped to head home Luke Ayling's cross to put the hosts ahead.
A 3-2 win, a long-standing rival sunk, a first win of 2020 secured. Crisis? What crisis?
NOW THIS IS A CRISIS: WEST BROM IN TURMOIL
Seven games without a win in the Championship for West Brom spells trouble - and the only saving grace is that the Baggies are still in the automatic promotion places.
Any hopes that the weekend win over West Ham in the FA Cup would prove a turning point were soon shattered at Cardiff.
Callum Paterson fired the Bluebirds ahead in the first minute of the second half, and despite Charlie Austin's penalty levelling up the match, Cardiff had the final say.
Lee Tomlin's free-kick was a sublime goal, curled over the West Brom wall and into the top left corner to clinch the points. Luton are next for West Brom, visiting the Hawthorns on Saturday.
FOREST EYES RETURN TO PROMISED LAND
It is worth remembering Nottingham Forest has not played Premier League football this century, having been relegated in 1999. Truth is, it has rarely looked like getting back, but maybe its time is approaching.
With the top two having looked suddenly vulnerable over recent weeks, Brentford and Forest have been among the teams making up ground at a rapid rate, which added spice to Tuesday's clash between the clubs.
Joe Lolley's strike made all the difference for Sabri Lamouchi's side, with the former Huddersfield player firing home with the help of a deflection in the 14th minute.
Forest last lost in the league at Huddersfield on 21 December, its seven-game unbeaten run raising real hope in Nottingham of a return to the top flight for the former European Cup winner.