White, who was making his Crucible theatre debut, overcame his fellow Welshman 10-6.
The 21-year-old admitted that he looked up to Williams and was pleased with the win.
"It means everything to me," White said. "I always looked up to Mark as a youngster growing up and it was a privilege to play him, and to play as well as I did means I'm over the moon. It's hard to take it all in at the moment but I'm delighted."
Breaks of 90 and 96 saw White over the line after the pair had previously been level at five frames all.
White could face former champion Stephen Maguire in the second round.