The Warwickshire professional, 23, seeded 89th for the tournament at the York Barbican, has lost three times in the final frame this season.
But he somehow regained his composure to win through 48-44 on the black.
"It's up there with my best wins, but I capitulated if I am honest. When it was 5-5 I fell to pieces. My arm went a bit like jelly," he told BBC Sport.
"I have lost a lot of close matches this year but, if you keep working hard, and I have been, then it pays off and I managed to scrape through. I am only in my third year a professional so you are always learning in these situations."
Wakelin, whose best ranking performance saw him reach the Indian Open earlier this year, apologised for his understandable show of excitement after sinking the final black to beat Welshman Stevens, the 2003 champion and former Masters winner.
"I had a bit of an outburst at the end and apologised to Matthew for that," said Wakelin, from Rugby. "But it means a lot to get through.
"I'm thrilled. He is such a tough opponent. He fought really hard to get back in to the match. He got a great clearance to go 5-5 and he put me under a lot of pressure. When you are playing against someone with such high calibre as Matthew you have to play well.
"Halfway through the final frame at 35-0 down with the balls all scrappy, it was difficult to see a way back. But I managed to chip away, play a couple of good shots and it came down to the pink. I messed up the black first time I played it and was lucky to get away with it."
Wakelin plays 25th seed Michael Holt in the second round on Sunday.
"Michael is a very tough player with a lot more experience than me so I will obviously be second favourite," added Wakelin. "But having won 6-5 on the black I am just grateful to still be in the tournament."
Chris Wakelin admits he "went to pieces" in his thrilling last-frame win over former champion Matthew Stevens.