Kevin McDonald scored twice for the Championship side to see off an Arsenal team made up almost entirely of young reserves and end a 25-year wait to reach the semi-finals of a major competition.
Stoke City were another Premier League casualty when they lost 1-0 to Derby County of the Championship for whom Nathan Ellington converted a last-gasp penalty.
In the remaining quarter-finals, holders Tottenham Hotspur are at Watford on Wednesday while Manchester United take on Blackburn Rovers at Old Trafford.
Burnley stunned Chelsea with a penalty shootout victory at Stamford Bridge in the previous round and also accounted for Premier League Fulham in this season's competition.
Arsenal manager Arsene Wenger stuck rigidly to his policy of fielding the reserves who thrashed Wigan Athletic and Sheffield United in earlier rounds, but this time the policy backfired with Nicklas Bendtner, one of only three regular first-team players in the side, guilty of some terrible finishing.
"We had six one-on-ones with the keeper and did not score," said Wenger. "If you have that many you have to question whether you can qualify for the next round.
"This team has a lot to learn. You have to be efficient from start to finish against a side like Burnley who were the best team we have come against in the cup this season."
McDonald opened the scoring after six minutes when he fired past Lukasz Fabianski after the keeper failed to hold a shot.
Arsenal replied with some great approach play and should have levelled when Bendtner was released on goal but dithered and shot straight at Burnley keeper Brian Jensen.
Bendtner wasted a similar opportunity before teenager Mark Randall was also denied by the inspired Jensen.
Fran Merida, a former Barcelona youth player, slalomed through the Burnley defence early in the second half but shaved the outside of the post with his shot.
Burnley doubled their lead after 56 minutes when Randall failed to clear the ball and McDonald pounced to poke a shot past Fabianski.
