After Mateusz Klich gave Marcelo Bielsa's side the lead on the stroke of half-time, Leeds punished the Clarets after the break as Jack Harrison added a second before claiming the two assists when Rodrigo came off the bench to score a brace.
The result gave Leeds a four-point cushion over 11th-placed Aston Villa before Dean Smith's side head to Crystal Palace on Sunday.
Nathan Tella scored his first goal for Southampton in their 3-1 win over already-relegated Fulham.
The 21-year-old came off the bench to net Saints' second, while Fulham's consolation came from 18-year-old Fabio Carvalho, his maiden goal for the Cottagers.
Che Adams scored a first-half opener for Saints, while substitute Theo Walcott restored their two-goal advantage with a clinical finish eight minutes from time.