Curran was the chief destroyer with 4-10 from his four overs, laying an early platform for the Stars to skittle the visitors for just 83 at the MCG on Tuesday night.
It was the third-lowest total of any BBL team at the famous venue and second-lowest recorded by the Strikers in the competition's 15 seasons.
In reply, Tom Rogers (32 off 40) top-scored as the Stars gradually chipped away at the meagre target and reached 4-86 with 29 balls to spare.
Captain Marcus Stoinis (23 off 20) also played a steady hand before he retired hurt with just two runs required after being hit on his right thumb.
He is set to have precautionary scans but expects to be given the green light to feature in the business end of the season, as well as the T20 World Cup.
"I reckon it's alright, yeah," Stoinis said.
"Initially it feels like nothing and after about 10 seconds you feel something and it's probably just not worth the risk at that stage of the game."
The result gave the Stars a 6-3 record, confirming their finals place, while Adelaide (3-6) will miss the top-four.
Fittingly, player-of-the-match Curran (9no off 4) hit the winning runs with a boundary.
Curran also dismissed four of the Strikers' top five batters as he posted the second-best figures of any bowler so far this season, and the best of his 256-game T20 career.
"Brilliant. He's been great for us all year," Stoinis said.
"He's bowling so well that I've pushed to just keep him on and bowl him four (overs) in that first 10 two games in a row.
"It's rare, but taking early wickets, we know how crucial that is in T20 cricket. So hats off to him."
Adelaide lost Alex Carey, Chris Lynn, Matt Short, Jason Sangha and one of their reviews in a disastrous first six overs, failing to hit a boundary as they were reduced to 4-21.
Liam Scott (18 off 20) eventually found the rope twice in the seventh over, with the Strikers managing three fours and two sixes in total.
Cameron Boyce (20 off 18) and Tabraiz Shamsi (8no off 22) put on 28 runs for the last wicket, but Adelaide's season crumbled in a woeful batting performance.
Stars legspinner Mitchell Swepson took 3-22 from his four overs and Stoinis claimed 2-8 from 3.3.
Boyce was struck on the helmet grille by a Haris Rauf delivery and carried on batting, but did not take the field after the change of innings.
The Stars could lock away a top-two spot with victory over Perth on Saturday, while the Strikers host the Melbourne Renegades in their last regular-season outing.
LOWEST ADELAIDE STRIKERS BBL TOTALS
68 v Melbourne Stars (15 Jan 2021)
* 83 v Melbourne Stars (13 Jan 2026)
87 v Sydney Sixers (10 Jan 2012)
88 v Perth Scorchers (26 Dec 2018)
9/90 v Melbourne Stars (9 Jan 2014)
LOWEST BBL TOTALS AT THE MCG
68 - Adelaide Strikers v Melbourne Stars (15 Jan 2021)
74 - Sydney Sixers v Melbourne Stars (10 Feb 2019)
* 83 - Adelaide Strikers v Melbourne Stars (13 Jan 2026)
9/90 - Adelaide Strikers v Melbourne Stars (9 Jan 2014)
99 - Melbourne Stars v Sydney Sixers (31 Jan 2020)
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