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
13. Martin Payero (sub) - 6
(for Crooks, 79) Had a few neat touches in midfield but Boro were defending most of the time he was on the pitch. 6 Photo: Robbie Jay Barratt - AMA
14. Aaron Connolly (sub) - 6
(for Sporar, 90) Played up alongside Watmore. Put himself about and saw a late chance saved by Dean Henderson. 6 Photo: Getty Images
15. Lee Peltier (sub) - 6
(for Taylor, 90) Helped shore up the left side of Boro’s defence in extra-time. 6 Photo: Alex Pantling
16. Sol Bamba - N/A
(for Jones, 118) Came on for the injured Jones in the closing stage of extra-time. N/A Photo: Stu Forster