The game finished 1-1 after extra-time before Boro won 8-7 in the shootout when United forward Anthony Elanga fired his spot kick over the bar.
Ralf Rangnick’s side missed the chance to open the scoring when Anfernee Dijksteel fouled Paul Pogba in the box, but Cristiano Ronaldo missed the subsequent penalty in the 20th minute.
The hosts did go ahead five minutes later, though, when Jadon Sancho’s deflected effort beat Boro keeper Joe Lumley.
United had more opportunities to extend their lead but were pegged back in the 64th minute when Matt Crooks bundled home an equaliser to force extra-time.
Here’s how each Boro player fared at Old Trafford:
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. MANCHESTER, ENGLAND - FEBRUARY 04: Matt Crooks of Middlesbrough is challenged by Paul Pogba of Manchester United during the Emirates FA Cup Fourth Round match between Manchester United and Middlesbrough at Old Trafford on February 04, 2022 in Manchester, England. (Photo by Clive Mason/Getty Images)
Matt Crooks. Photo: Clive Mason
. Joe Lumley - 6
Almost gifted Bruno Fernandes a goal in the second half but also made some important saves when he was quick off his line. 6 Photo: Robbie Jay Barratt - AMA
. Anfernee Dijksteel - 6
Recovered after conceding an early penalty. Gained more confidence when stepping out of defence with the ball. 6 Photo: Getty Images
5. Isaiah Jones - 6
Struggled to deal with Sancho in the first half but managed to get forward more after the break. Played a key role in the equaliser. 6 Photo: Getty Images
6. Neil Taylor - 6
Took up some advanced positions early on but began to get pinned back as Boro invited pressure in the first half. 6 Photo: Getty Images
7. Jonny Howson - 8
Covered plenty of ground and positioned himself well to win the ball back and retrieve possession. 8 Photo: Robbie Jay Barratt - AMA
8. Matt Crooks - 7
Conceded possession for the opening goal but worked tirelessly before he was substituted on 79 minutes. Was rewarded when he arrived late in the box to equalise. 7 Photo: Robbie Jay Barratt - AMA