Robertson notched a 106 break as he roared into a 4-1 lead, with Higgins mustering a solitary point in frames three, four and five combined.
Neil Robertson wins second UK title boosted by stunning 147 break
But the Scotsman was not finished. Two monster breaks of 134 hauled him back into contention and after 10 frames the score was balanced at 5-5.
Could Higgins complete a remarkable comeback?
It looked a real possibility when he sunk a long red, but his follow-up green found the far jaw as Robertson returned to the table to wrap up a nervy win.
John Higgins and Neil Robertson during their quarter final
Image credit: Reuters
“The match was an emotional roller-coaster,” said Robertson. “I played really well to go 4-1 up and then John played like a champion to come back. He had chances in the last frame and I just had to be ready in case he missed. I was relieved to get over the line.
I have won everything in the game so now I have to motivate myself to win things again and build my legacy.
The match was briefly brought to a halt when a rogue moth landed on the table. It initially evaded the clutches of referee Jan Verhaas, before he caught it at the second attempt.
It was equally anxious in Friday afternoon's other quarter-final, with Liang Wenbo edging Marco Fu 6-5.
Liang started in sensational fashion as he reeled off three centuries – 106, 132 and 138 – but Fu eventually got amongst the action, battling back from 5-2 behind to force a decider.
That was as good as it got for the Hong Kong potter, however, as Liang came through to take the 11th frame.
Liang Wenbo celebrates after winning his quarter final
Image credit: Eurosport
“I made a great start with three centuries, then he put me under a lot of pressure from 5-2,” said Liang. "When it got to 5-5 it was very tough.
I thought I would lose but I just tried to stay calm and enjoy it. Marco and I are friends and it was a good game.
In the evening session, Mark Selby blitzed through his quarter-final 6-1 against Matthew Selt.
The Englishman posted four breaks over 50 during his serene passage, with Selt only managing to win the third frame.
Neil Robertson 6-5 John Higgins
Liang Wenbo 6-5 Marco Fu
Mark Selby 6-1 Matthew Selt
Martin Gould 5-6 David Grace
Robertson whitewashes Selby to set up final meeting with Liang Wenbo