The Australian reached the three-figure milestone in the 22nd frame to level at 11-11 - the first time he had been on equal terms since the match was in its infancy at 1-1.
His historic century came in the middle of a run of five straight frames and, having trailed 11-8, Robertson stormed back to snatch a sensational victory from the jaws of defeat.
The haul of 101 was Robertson's seventh century of the tournament and was celebrated wildly by the world number one along with the knowledgeable crowd at the Sheffield venue.
The 2010 world champion moves into the semi-finals and a date with Mark Selby.
"It was great that it came at one of the most crucial times of the match," said Robertson. "I wasn't even thinking about the century until there were only a few balls left and the frame was finished - then I really went for it.
"I played a good shot on the last red, and then I couldn't miss the green. I would rather make the century of centuries here than a 147. I've made a couple of 147s and it's nowhere near the same achievement.
"Nobody will ever achieve 200 centuries - that would be impossible. For me to be the first player to achieve 100 centuries in a single season is a great honour. It may raise the bar for break-building - the players might try a bit harder when they're on 70 or 80.
"Stephen Hendry was the one who always tried to clear up no matter what, and that was the approach that I've taken all season."
Robertson's upcoming clash with Selby will be a repeat of the UK Championship final, which the Australian won 10-7.
"Best of 33 with Mark Selby is going to be really tough," Robertson added. "I can't allow myself to get bogged down so I have to approach the match the same way I did tonight. When I'm playing really well, I'm very difficult to beat."
A disappointed Trump said: "Neil making 100 centuries doesn't mean anything to me. I congratulated him at the end of the match - that's all I could do. He likes to show his emotions and the crowd can enjoy that sometimes, so I've got nothing against him doing that.
"If he plays like that, he'll have a good chance. If he gets to the final, Ronnie [O'Sullivan] won't enjoy playing against him."
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