The Cottagers took the lead after just four minutes as Zoltan Gera hooked the ball home after Bobby Zamora had nodded back Paul Konchesky's cross.
Chelsea had to wait until deep into the second half before drawing level.
The Blues ground out a 1-0 win thanks to a goal early in the second half against a disappointing Stoke outfit.
Black Cats old boy El Hadji Diouf denied his former club a first win in seven games as Blackburn twice came from behind.
Sunderland grabbed the lead in the 52nd minute after Darren Bent volleyed home.
Within a minute Rovers were level as Morten Gamst Pedersen hit a low right-footed shot which went in off the post.
Bent then claimed his second on 65 minutes only for Diouf to head an equaliser with 13 minutes remaining.
Two late goals proved decisive as the Clarets miserable away form continues.
Substitute James Vaughan scored on 83 minutes and Steven Pienaar doubled the advantage in the final minute.
Tottenham boosted their Champions League hopes, and kept a fourth clean sheet in a row in the process, after a 2-0 home success over London rivals West Ham.