Robertson needed two snookers but was only able to force one from a delicate exchange which revolved around that black ball, the Australian having nursed a red alongside it to open up the possibility of Williams potting the black.
Williams produced one of the shots of the tournament to avoid doing so, moving the red in a length-of-the-table escape without touching the black, and had the match in the palm of his hand as Robertson set about finding another escape route.
Robertson set what seemed a routing snooker from which Williams would escape, but he chose to swerve and got the shot all wrong, not only fouling but leaving his opponent on the yellow.
Robertson needed the colours to win and he somehow held his nerve to clear them in a tense and exhausting finale, his pot on the green particularly impressive after a poor positional shot.
Asked to explain how he'd pulled it off, an emotional Robertson said: "I don't know, just never give up. Never, ever, ever give up. Any kids out there watching, doesn't matter how it looks.
"It's something so special about both of us coming out to a standing ovation (ahead of the deciding frame).
"To have a match that finishes like that, you'll probably never see that ever again in the sport. It's going to take some hours for that to sink in.
Asked if this was the comeback of his career, he added: "It has to be, I needed two snookers at 5-5. Mark played an amazing shot where he's clipped the red away. I played a really good snooker on the yellow.
"The green is one of the best pressure balls I've potted in my career. Maybe sometimes I would've been thinking about playing safe, but if you need two snookers and you've got the chance to go for it, you have to go for it. I did, and managed to hold myself together at the end."
Robertson returns on Sunday looking to win the Masters title for a second time, 10 years on from his first.
"I almost feel I've got nothing to lose," he said. "I was out of the tournament. It might make me a dangerous man tomorrow. I just can't wait to play in front of this amazing crowd again."
Williams said: "Could've, should've... it's all part of the game. If the black goes in on 67, you don't come back to the table. It wobbled (and) gave me 25 minutes of pain.
"I probably should've come off two cushions (on the green). I've got no problems with that, I've lost in the semi-final, and best of luck to Neil.
"I've loved every minute of it. The crowd are fantastic, the arena's fantastic. Unfortunately I'm not in the final tomorrow but best of luck to Neil... I say (that), I don't care who wins it, I won't be watching!
"I tried my guts out, it was a great match."
Hawk downs Trump
Barry Hawkins held firm to beat Judd Trump and secure a second crack at the Masters final, five years on from a 10-1 defeat at the hands of Ronnie O'Sullivan.
Trailing 5-4, Hawkins looked to have missed an opportunity as Trump rallied to take control, but the 42-year-old kept Trump from scoring thereafter with breaks of 76 and 58 enough to cross the line.
"I'm speechless," said Hawkins. "I was so nervous at the end. I can't believe it, I'm so pleased."
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