Otis Khan’s 29th-minute spot-kick was enough for the Glovers who progress to the second stage of the competition for the fourth time in five attempts.
Yeovil boss Way was forced to make two changes for the tie and was delighted with the players who stepped in as well as the rest of the team.
“I’m very happy,” he told iFollow Glovers.
“We had a lot going against us with our two centre forwards missing and Surridge having to play the lead role with Browney and Khan either side.
“I thought Jake Gray covered every blade of grass and played a massive part. Everyone played a massive part but to come in and play like he did was great – same goes to Rhys Browne.
“Omar Sowunmi was very effective at the back, is growing stronger every game and has a good partnership with (Nathan) Smithy.
“It shows what we’re capable of. You can explain the success stories we’ve had in the competition but when they cross the white line in the arena it’s up to them and a lot of credit goes to the players for the performance they put in.”
Way added: “It’s also good for the football club, directors, supporters and all the backroom staff – they deserve to get through to the next round for all the hard work they’ve been putting in.
“Fingers crossed we can get a good tie in the next round and create a little bit of history.”