Like Williams, who came from behind to beat Ronnie O'Sullivan 6-5, Lisowski had to fight back to edge out Vafaei in a high-quality encounter.
The world number 12 opened in style as runs of 96 and 76 carried him into a 2-0 lead but Vafaei hit back strongly to win the next three, equalling Williams' tournament-highest break of 143 in the process.
Lisowski levelled after a break of 67 but the next three frames were tightly fought.
Lisowski eventually inched 5-4 ahead and then closed out victory with a 74.
"It's like a dream," he said. "Three or four years ago I couldn't win a game here but now to be in the semi-finals feels amazing."