The Strikers' 8-165 looked about 20 runs short on Tuesday night at Adelaide Oval but Jamie Overton, Hassan Ali and Lloyd Pope all contributed with the ball to secure the win.
After captain Matt Short took a crucial diving catch to remove David Wiley in the second-last over, Luke Wood dropped an easy chance off Nathan McAndrew.
But Wood redeemed himself, bowling a perfect last over.
Sydney needed 13 to win, but Wood restricted them and they finished on 7-159.
Sydney captain David Warner, unbeaten on a match-high 67, smashed his bat into the ground in frustration when it was clear in the last over that his side would not win.
While Adelaide are in the finals hunt, the Thunder are last with only one win from seven games.
Warner and Matt Gilkes took the Thunder to 0-73 in the 10th over and it looked as though Sydney would cruise to the win.
But Gilkes' dismissal for 43, caught behind off Overton, triggered a top-order wobble.
Sam Billings and Sam Konstas went cheaply in a 3-7 spell that had the Thunder suddenly stalled at 3-77 in the 12th.
Warner and NIc Maddinson then combined for a 44-run stand as the momentum shifted again.
After Overton dismissed Maddinson for 21, Pope bowled Daniel Sams and Chris Green was run out by an excellent throw from Jerssis Wadia to leave the game in the balance again.
Overton was the pick of the attack with 3-25 from his four overs.
After his 3-30 BBL debut on Sunday, South African spinner Tabraiz Shamsi was left out of the Strikers line-up for Hassan.
Chris Lynn hit a towering six to open Adelaide's innings after Warner won the toss. But Lynn and fellow opener Matt Short did not survive the first two overs.
Liam Scott was unbeaten on 49, but was kept at the non-strikers' end for much of the final overs, while Mackenzie Harvey contributed 38.
Wes Agar was outstanding at the death, taking two wickets in Adelaide's last over.
He snared 3-16 from three overs, while Tanveer Sangha's four overs only cost a miserly 0-19.
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