With the game finishing 0-0 after 90 minutes, Darren Fletcher's side needed an additional 30 minutes to see the job over the line.
An own goal from Luca Crolla got the ball rolling, before Chido Obi added a second for United, shortly after the opener in M16.
Max Nessling scored for the Rams in the second period of extra time, but we hung on to confirm victory in front of 2223 supporters.
FIRST HALF
It was a tense opening for both teams at the Theatre of Dreams, with a midfield battle ensuing early on.