The veteran cuemen, who between them have won seven world titles, resumed Wednesday all square at 8-8 in the best-of-25-frames quarter-final at the Crucible Theatre in Sheffield.
Wales star Williams then moved to the brink of victory by winning the day's opening four frames to go 12-8 ahead, but four-time champion Higgins ensured the match would go to a deciding frame after the Scotsman reeled off breaks of 94, 114 and 67.
Yet it was Williams who prevailed after Higgins missed a tense blue on the top cushion to seal a 13-12 victory.
"There was not one bit of nerves," Williams, 50, insisted to the BBC afterwards. "I don't really get it. I've only had nerves a handful of times in my career."
He added: "I wasn't thinking he (Higgins) would miss it. From 12-8 to 12-all I had not done much wrong and I was thinking 'What can I do?'
"It's just a pleasure playing now. We got the standing ovation walking in and everyone was cheering. You have to enjoy it because you don't know how many times you'll be coming back here."
Ronnie O'Sullivan, meanwhile, kept China's Si Jiahui at bay to lead 10-6 heading into their final session later Wednesday, with the Englishman bidding to win a modern-day record eighth world title.
No Chinese player has yet been crowned snooker world champion but Zhao Xintong moved a step closer by booking his place in the last four for the first time after he won the one frame he needed against Chris Wakelin to turn a 12-4 overnight lead into a 13-5 victory.