Rankin (0 handicap) made a 33 break with his third visit of the opening frame to move 33-15 ahead.
Lee (15 handicap) replied with 14 points, but Rankin potted a further three blues to be 51-29 in frront, then added eight more points.
Lee took advantage of two careless safety shots to close to within eight points, but erred on the yellow. Rankin potted yellow, green and brown, but failed on the frame-ball blue.
With the blue over the middle, Lee had an easy pot but he then missed a straight pink with the black available.
He was given another chance and potted both colours to snatch the frame 69-68.
Rankin went 60-23 ahead in the second frame, helped by breaks of 31 and 19, but Lee replied with 15 and 12 to trail 60-50.
Rankin gained seven points with the last red and then fluked the yellow before taking advantage of a mistake to clear the remaining colours for a 25 break, taking the frame 94-50.
Lee went 25-0 ahead in the third frame, only for Rankin to make a 23 break to move 44-33. Lee looked favourite as he built his score to lead 47-44 when he potted the brown.
However, another mistake allowed Rankin to take blue and pink to win the frame 55-47.
The fourth frame was a dour affair, with Lee potting six reds but only adding 12 points to his handicap, while Rankin sank eight reds and colours to lead 43-27.
A poor safety by Rankin let Lee in to add a black to the last red, but he failed on a simple yellow with a clearance on.
Rankin cleared to the pink to lead 57-35 and leave Lee needing two snookers. He got one, but eventually Rankin potted the pink to win 63-41.
In the semi-finals, Rankin defeated Malton's Lee Taylor and Lee ousted Bootham's Paul Mahoney.
Heworth's Clive Whyte took the high break prize for his run of 53.