Fred and Anthony Elanga scored to put the team 4-2 away after losing a two-goal lead in the early stages of the second half.
In a searing atmosphere, the visitors took the lead in the 34th minute, with captain Harry Maguire heading home to score his side’s first league goal from a corner all season.
As the rain grew more torrential, United added to the home side’s woes when Bruno Fernandes, who scored a hat-trick in the reverse game earlier this season, poked another header into the net in stoppage time of the first half.
Seemingly dead and buried, Leeds turned the game around with two goals in under a minute at the start of the second to lift the roof off Elland Road.
Spain striker Rodrigo’s cross looped past United striker David de Gea and into the net before Dan James slotted in Leeds substitute Raphinha to equalize at the far post.
The visitors were the ones to cross the line, however, as Fred put his side ahead with 20 minutes left before Elanga settled the game with a late fourth.
The result moved fourth-placed United to 46 points, four behind Chelsea in third, while Leeds remain 15th, five points above the relegation zone.
Additional Reuters reports