The home side came from a goal behind to force the shootout after David McGoldrick opened the scoring in the third minute.
Burnley lost Johann Berg Gudmundsson on the quarter-of-an-hour mark as the Icelandic international was left clutching his knee following a challenge from Jack Robinson.
Matej Vydra equalised in the 67th minute when lifting the ball over Wes Foderingham from Jay Rodriguez's cushioned header before the tie went to a shootout.
Chris Wood, Vydra, Josh Brownhill and Erik Pieters all converted before Robbie Brady kept his cool to wrap up the victory.
Matej Vydra equalised in the 67th minute when lifting the ball over Wes Foderingham from Jay Rodriguez's cushioned header before the tie went to a shootout.