Williams and Higgins, 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 reach the semi-finals with a 13-9 win.
The Englishman is bidding to win a modern-day record eighth world title and led 10-6 heading into their final session later on Wednesday.
O'Sullivan lost three of the opening four frames but took the 21st and 22nd to seal a hard-fought win.
"I have give it my best but I am hanging in there. I would like to be playing better but I have to try and find some form at some point," he told the BBC.
"I don't feel rusty, I have only been playing for a month but I just couldn't face playing. It gets harder, look at the way Judd is playing.
"I won't be able to get away with what I have done so far. I can't get any worse, put it that way.
"I'm just grinding it out, I had some work done on my cue today, I am playing awful and my cue feels awful but I am still here.
"I miss the buzz of playing well. He had lots of chances, I was trying everything and he let me off the hook.
"Each match is an opportunity. The Chinese are such nice boys I love to see them playing well," he added.
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.
World Snooker Championship semi-finals
Ronnie O'Sullivan v Zhao Xintong, Thursday May 1st, 13:00 BST
Judd Trump v Mark Williams, Thursday May 1st, 19:00 BST