Robertson won the 2007 tournament but his chances of victory this year nearly slipped away from him.
The world number three originally trailed 2-0 but he won the third frame after a foul by Allen. He then won five consecutive frames to lead 5-2 and withstood a comeback by Allen to reach the final.
The 34-year-old will now play the winner of Ronnie O'Sullivan and Joe Perry on Sunday and he admitted after his win that Allen's mistake was crucial.
"The first part of the match I wasn't quite with it, making a couple of half-awkward mistakes and letting Mark in," he told BBC Sport.
"The turning point was the foul and then scoring 65 in the third frame, otherwise I would have gone 3-0 down. After the interval I came out and played brilliantly."