The tie was taken into injury-time before United eventually won 3-2.
Former Arsenal striker Chido Obi didn't make the United squad on the night, hinting he will be involved in the senior FA Cup fifth round tie against Fulham on Sunday.
Jack Fletcher had United ahead before Louis Copley drew Arsenal level just before halftime. The Gunners then took the lead as Max Dowman won and converted a penalty.
United drew level six minutes before normal time through fullback Jaydon Kamason's drive.
Then in extra-time, United winger Bendito Mantato found the winner after good lead-up play from fellow substitutes Jim Thwaites and Samuel Lusale. The tie concluded with Arsenal defender Will Sweet seeing red.
United reach the semifinal for the first time since 2022, when a team including Kobbie Mainoo and Alejandro Garnacho won the competition.