After two sessions of their match, Hawkins led 10-6, but Williams, a champion in 2000, 2003 and 2018, was unable to fight back as the six frames went alternate ways, with Hawkins securing the victory.
Eleventh-seed Hawkins, who turned 47 on Wednesday, was helped by two breaks of 85 – one in frame 18 and one in frame 20 – to reach the Crucible quarter-finals for the seventh time.
“I just had to try my best over the three sessions. In two I played OK but the second session, I didn’t play very well and that’s when he got a good lead,” said Williams.
“I tried to come back but he was just the better player. I just don’t make enough breaks any more. I broke down on 40 or 50.
“It was the best reception I’ve ever seen here. They were all on their feet. It probably won’t happen again with all three of us [of the ‘Class of 92’] in the same session.
“There’s nothing else I can really do, I can’t practise for hours each day and I can’t get better – as you go on, you get worse and worse.”
Hawkins, who will play Northern Ireland’s Mark Allen in the quarter-finals on Tuesday and Wednesday, said: “This is definitely up there [as his best Crucible experience]. It’s not easy beating someone like that to get to the quarter-finals.
“I didn’t just fall over the line so I’m quite happy with the way I played and my game – I enjoyed every minute of it.
“I used to have too much respect and would crumble against these sorts of players but over the last four or five years I’ve got more belief.”
On the standing ovation at the start of the session, he said: “They were not standing for me, they were standing for the Class of 92 but when I walked out, what a feeling. It’s the best place to play when it’s like that.”