World number four Bunting came from 6-4 down to win 12-10, averaging 108 over the final stretch (92.71 overall) and fending off a late surge from the Scotsman - including a tie-break leg - to progress to the next round.
"I was so so nervous playing a legend of the sport on this stage, I tried to put it to the back of my mind, but during the game I just couldn't," Bunting told Sky Sports after the match.
"I'm happy with the win but I need to just get to bed - I'm shattered.
"I've got massive respect for Gary. He's a quality operator, he's one of the best players to grace the stages.
"What a win in my career. We go again.
"I felt nervous anyway. I had Theo, Toby, the Mrs with us tonight, but we're through and I can relax now."
Wade waltzes through
Earlier, James Wade maintained his impressive form with an 11-5 win over Wessel Nijman.
The ninth seed produced a 128 to leave 140, then hit 100 to set up double top – which he found at the first attempt.
The impressive victory secures another place in the Matchplay quarter-finals for him, with an average of 102.5.
Gian van Veen extended his impressive debut campaign with a commanding 11‑5 win over fellow Dutchman Danny Noppert in the night's opening fixture.
Van Veen broke early to lead 7‑2 and never looked back, averaging 94.94 and landing an impressive 44 per cent (11/25) of his checkouts to set up a last-eight tie with Wade.
Clayton in control
Jonny Clayton completed the evening's list of quarter-finalists with an 11-9 win against Belgium's Mike De Decker, sealing victory with a bullseye finish after a commanding display at the oche.
"It wasn't experience at all," Clayton said after the victory, which means he'll face Bunting in the next round.
"Nobody has thrown near to what we can, I struggled big time, Mike struggled big time.
"It's one of those games I glad to get over the line.
"I just think I was thinking too much about the game, I know how good Mike is.
"I didn't play with a smile on my face, I was overthinking things. I couldn't find anything today. Hopefully on Thursday I can do better.
"I'm here to win the tournament. I'm a proud Welshman, I'm here for the fans."
The second-round matches continue on Wednesday with Andrew Gilding against Dirk van Duijvenbode before Chris Dobey faces Gerwyn Price, Luke Littler plays Jermaine Wattimena and Michael van Gerwen takes on Josh Rock.