With star international pair Harmanpreet Kaur and Stafanie Taylor at the crease and the Thunder requiring nine to win off the last over to reel in the Scorchers' 7-127, the odds appeared in the visitors' favour.
But Graham (2-8 off three overs), the WBBL's leading wicket-taker this summer, had ice in her veins.
She immediately dismissed Kaur, who battled for rhythm all day, scoring 19 off 31 balls, without a solitary boundary.
Then, with five needed off the last ball, Taylor could only squeeze Graham's perfect yorker for a single, setting off wild celebrations at Lilac Hill from the Perth camp after sweeping the Thunder 2-0 for the weekend.
Opener Rachel Priest (0) and No.3 Naomi Stalenberg (four) both holed out early, trying to slog Nicole Bolton (2-26) before Thunder skipper Alex Blackwell (37) and Ashes Australian captain Rachael Haynes (33) steadied the ship.
But when the experienced pair departed, the asking rate escalated and neither Kaur nor Taylor was able to make leeway against some smart bowling.
Earlier, the Scorchers, minus gun opener Elyse VIllani (hamstring), slipped to 3-30 after the powerplay, which soon became 4-44.
Bolton, who enjoyed a fine all-round game and piloted the recovery mission, top-scored with a sprightly 50 and put on 78 for the fifth wicket with Lauren Ebsary (33) to give the hosts something decent to defend.
The Scorchers sit in third place on the WBBL|04 ladder, and will enjoy a week off before taking on league leaders Sydney Sixers back at Lilac Hill next Sunday.