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.
1. Nick Pope 6
Left exposed for Sheffield United's early opener, but was rarely tested thereafter. Comfortably denied McGoldrick after the striker's opener and did well to punch clear Norwood's set-piece when under pressure. Made a big save to deny McBurnie late on..
2. Matt Lowton 6
Lowe exploited the space in behind the full back to deliver for McGoldrick's opener, which was an area targeted by the Blades on a number of occasions. Delivered a sumptuous cross for Rodriguez and tightened up significantly in the second half.
3. Jimmy Dunne 6
Beaten to Lowe's cross by McGoldrick for the opener and, understandably, showed a few nerves, particularly with his distribution, However, once settled, he defended superbly, held his line well and was a real force in the air.
4. Kevin Long 7
Barely put a foot wrong, stood his ground well, prevented his marker from turning when the ball was played into him and really helped Dunne through the tie. Always alert inside the box and made a big block to deny McBurnie late on.