Naby Keita pulled a goal back before the hour mark, but Uth scored his second goal with a header before midfielder Lukas Rupp hit Hoffenheim’s fifth.
England Under-21 international Ademola Lookman, on loan from Everton, set up Dayot Upamecano to score Leipzig’s second on 88 minutes.
The result leaves Hoffenheim fifth, just two points from the Champions League places, while six-placed Leipzig are four adrift.
? #TSG, is the finest team, the world has ever seen!! (74′)
— TSG Hoffenheim EN (@achtzehn99_en) April 21, 2018
Mid-table Hertha Berlin romped to a 3-0 win at German Cup finalists Eintracht Frankfurt as Davie Selke, Mathew Leckie and Alexander Esswein scored.
Frankfurt suffered a cup hangover, after beating Schalke in Wednesday’s semi-final, and remain seventh.
Club captain Christian Gentner and Berkay Ozcan scored Stuttgart’s goals in their 2-0 victory over Werder Bremen.
Ex-Tottenham Hotspur midfielder Lewis Holtby boosted Hamburg’s hopes of avoiding relegation in their 1-0 win at home to Freiburg.
Holtby jinked through the Freiburg defence and tapped home just after the break to leave second-from-bottom Hamburg five points from safety with three games left.
Hamburg are the only current Bundesliga to have played in every season since its foundation in 1963.