Sean Dyche's side will now entertain the winners of the third round tie between holders Manchester City and AFC Bournemouth, which takes place on Thursday evening.
Burnley, who fielded three debutants including Bailey Peacock-Farrell, Bobby Thomas and Josh Benson, scored in either half to seal their progression in the competition.
Josh Brownhill opened his account for the club with a thunderbolt from 25 yards out before Matej Vydra netted with a cool finish deep into stoppage time.
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Josh Brownhill opened his account for the club with a thunderbolt from 25 yards out before Matej Vydra netted with a cool finish deep into stoppage time.