Otis Khan’s spot-kick was cancelled out by a late Marcus Harness curler which forced the tie at Huish Park into extra-time.
Vale had been reduced to nine-men by this point and Khan hit a stunning shot in off the post to put the hosts back in front five minutes into the additional period.
But another visiting wonder strike from Antony Kay restored parity once more before Francois Zoko’s close range effort saw the Glovers advance and set up a match with Sky Bet League One outfit Bradford City.
“That’s the magic of the FA Cup – Paul Daniels couldn’t have even made that up!” Way told iFollow Glovers.
“It was entertaining and the supporters got their money’s worth that’s for sure.
“Obviously we made hard work of it but I remember being in that situation at Reading away in the Championship where I think we were down to eight players and held on for a draw.
“To get through to the next round is very important for the football club financially and I think it’s the fourth or fifth time we’ve got to the third round since I’ve been here.
“It’s a great achievement for the players and I want to make sure the respect is shown for the effort they put in. Yes, we did make hard work of it but it’s important we’re through.
“A big well done to the players and we look forward to Bradford now.”