Plymouth levelled through Rommy Boco after Reuben Reid flicked on a long clearance from Cole and he lifted the ball over Scott Brown.
Less than a minute later, Argyle took the lead when Andre Blackman's run and cross was met by Lewis Alessandra at the far post. His initial shot was blocked by Sido Jombati, but he followed up to tap in the rebound.
Plymouth extended their lead in the 57th minute when Conor Hourihane's free-kick was headed in by Hamza Bencherif.
The home side were handed a lifeline on the hour when Gillespie was tripped by Nelson in the box and the full-back was given a straight red. Gillespie saw his spot kick saved by Cole, but he followed up to score his second of the match.
Six minutes later Jason Taylor levelled with a deflected shot to take the tie to penalties. Terry Gornell's spot kick was saved and Plymouth scored all five of theirs to progress to the second round, with Marvin Morgan hitting the decisive fifth.