Rovers took a slender 1-0 lead from the first leg to Turf Moor and after a goalless first half, the visitors took the lead through Hugo Fernandez in the 55th minute.
Three minutes later and it was 2-0, this time Ryan Edwards struck from a long range free kick.
Burnley pulled a goal back from the penalty spot with just five minutes left to play, Steven Hewitt firing home the spot kick.
But that wrapped up the scoring and Blackburn won the game 2-1, 3-1 on aggregate.
Rovers will meet the winners of Manchester United v Chelsea in a two-legged final.