Needing 17 runs off the last of their allotted 20 overs, Graham crashed 16 from the bowling of Heather Knight, including two off the final ball, to haul her side level.
The visitors then took home the points by scoring 16 off the additional over, while the Hurricanes could only manage 2-10.
Nicole Bolton thumped 15 off five balls in the added over, including a massive six.
The Scorchers had seemed dead and buried heading into the final stages of the match, with 52 required off the last five overs, but the 22-year-old Graham picked the perfect time to post a season-best 44 not out off 30 deliveries.
"That was unreal," Graham said.
"I didn't know I was bowling the super over until about two minutes before we went on, I had to run down and change back into my spikes.
"I had to back my self when I was handed the ball, I knew what I needed to execute and if I did that I knew they wouldn't be able to score the runs."
Lauren Ebsary (20 off 11 balls) played a crucial supporting role with Graham, as the Scorchers eventually reached 6-173.
The Scorchers, still without injured captain Mel Lanning, were well placed at the midway point of their innings, only for the required run-rate to reach a seemingly insurmountable level.
In-form opener Elyse Villani hit 41 off 34 balls before Hurricanes' skipper Sasha Moloney (2-26) claimed the prized wicket.
The win sees the Scorchers move into the top four with three wins and three losses.
The team now travel back to Perth to gear up for a double header against the Melbourne Renegades at the WACA Ground this weekend.