Higgins had lost his previous seven encounters with Trump and, after a ragged opening spell littered with uncharacteristic errors, appeared in danger of being swept aside.
Trump surged into a three-frame lead with calm authority, yet there was an underlying fragility to his advantage.
The match pivoted in the fourth frame. Trump passed up two clear chances to extend his lead, and Higgins seized the opening with the urgency of a man sensing escape.
Frame five was settled by a superb 104 break, a reminder that fluency and nerve remain very much intact at 50.
Trump continued to find ways to edge ahead, capitalising on missed reds to steal the sixth and eighth frames with half-centuries, but Higgins refused to retreat. A composed 57 clearance secured frame seven, keeping the contest alive and the atmosphere inside Alexandra Palace taut.
At 5-3, Trump seemed poised to close the door, but instead, the semi-final unravelled in astonishing fashion.
In the ninth frame, with victory in sight and only four balls remaining, Trump missed a straightforward brown that was punished by his seasoned opponent.
Breaks of 70 and 57 then carried Higgins through the closing stages as he reeled off three consecutive frames, each one tighter than the last, to complete a turnaround that ranks among the finest of his Masters appearances.
In doing so, he became the oldest player ever to reach the final of a Triple Crown event, another entry added to a career that refuses to fade quietly.
Awaiting Higgins in Sunday's final will be either Kyren Wilson or Wu Yize, who contest the second semi-final from 20:00 CET on Saturday.