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Robertson recorded three century breaks as the Australian took a 5-3 lead but Scotsman Higgins roared back to level the match in front of a home crowd.
The semi-final came down to the final ball, Robertson holding his nerve as Higgins conceded on the black.
Robertson will now face either Ding Junhui or Mark Williams in the final.
"I can't remember winning a decider on the black for years and years," Robertson told BBC Sport.
"To play the way I did throughout the whole match, against the world champion - I'm going to watch the video of that match a hundred times."
His early break of 114 set the tone of the opening frames, a venomous shot into the pack sending the 27-year-old on his way as both players produced a very high standard of snooker.
Higgins never let Robertson get too far ahead, however, and the defending champion hauled his way back to a 3-2 lead with a well-executed plant in a scrappier fifth frame.
Once again, a brave shot to split the remaining reds earned Robertson the sixth frame with a break of 130 and both players squandered opportunities to seize the initiative in the seventh.
Higgins looked to finally find the momentum as he crafted an excellent pot, squeezing a red inside the blue guarding the yellow pocket, and fell kindly on to the black.
But the Scotsman subsequently positioned the cue ball horribly, allowing Robertson to take his chance, and he pulled two frames clear with a break of 128 in the eighth - his 99th competitive century.
Higgins fought back after Robertson had posted a break of 49 in the ninth frame, producing a frame-winning clearance, then dominated the 10th after Robertson struggled to extricate himself from a snooker.
That took the tie to a deciding and attritional 11th frame, in which Robertson spent many a minute poring over a succession of challenging shots.
The pressure told as Robertson, forced into an unsuccessful attempt at a tricky red, left the path clear for Higgins.
However, Higgins then had trouble of his own, making a tough red but watching in horror as the pink also dropped.
The pair traded snookers with only the colours remaining, Higgins seven ahead of Robertson, before Higgins inadvertently potted the yellow, snookering himself in the process.
Robertson missed a chance to cut the green into the middle pocket and Higgins then squandered a comparatively simple brown under intense pressure.
The frame came down to the black, Higgins duly conceding having left it over the pocket after an ultra-fine shot from distance.
"I don't think many people in the world are capable of that comeback," said Robertson.
"John and Ronnie O'Sullivan are maybe the only two at the moment."
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Neil Robertson survived a fightback from John Higgins to earn a superb 6-5 victory in an intense match and reach the final of the Grand Prix in Glasgow.