Steve Bruce’s side should have been home and dry in normal as they had the Clarets defence on the ropes.
However, Sean Dyche’s side held out for spot-kicks, where they emerged victorious after goalkeeper Wayne Hennessey saved Allan Saint-Maximin and Miguel Almiron’s efforts.
Here’s how we rated each United player as the club’s long wait for a major trophy continues:
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9. Ryan Fraser - 5
Had his moments but struggled to get into the match. A central midfield role does not suit him. His best bits followed when he drifted out to the left. Photo: Ian MacNicol
10. Joelinton - 6
Made some decent runs along the Burnley backline but as ever with Joelinton, he just lacked that urgency and the attacking knowhow. Scored a great penalty however. Photo: Michael Regan
11. Dwight Gayle - 5
Denied a goal by the underside of the crossbar but apart from that, it was another quiet outing for Gayle. Photo: Michael Regan
12. Joe Willock (on for Manquillo, 75) - 6
Introduced with 15 minutes remaining. Scored his penalty in the shootout. Photo: LINDSEY PARNABY