The Strikers were in trouble after being bowled out for 135 before Rashid (4/22) intervened and turned the match Adelaide's way during the Sixers’ chase.
Rashid picked up James Vince, Jack Edwards and Jordan Silk off successive deliveries for his third career Twenty20 hat-trick, Adelaide’s first as a club and the fifth in BBL history.
The lethal leg-spinner celebrated wildly with teammates after completing the trifecta with a perfect googly which bamboozled Silk (16) and crashed into his woodwork as the Sixers crashed from 3/79 to 6/83.
But unlikely batting hero Josh Hazlewood, on his 29th birthday and with 11 required for a Sydney Sixers victory, crunched Peter Siddle for three straight boundaries in the penultimate over for a thrilling two-wicket win with eight balls to spare.
The result lifts the Sixers into top spot on the tournament table while the Strikers sit third with three wins, three losses and a no result.
The Sixers' chase started unconvincingly when openers Justin Avendano (11) and Josh Philippe (eight), Sunday's match-winner at Coffs Harbour, holed out in Michael Neser's first over as he claimed his 50th wicket for the club.
When Player of the Match Tom Curran (21) was yorked by Harry Conway, the Sixers looked shot before Hazlewood, who entered the match with five career BBL runs, struck an unbeaten 12 off three.
Earlier, Curran's career-best T20 haul of 4/22 left the Strikers reeling, the home side losing 5/10 to slump from 3/110 to 8/120 and losing their last seven wickets for 25.
Jake Weatherald (47) and Alex Carey (32) led the way for the Strikers with the bat but lacked support as wickets fell regularly.
The Strikers will regroup to tackle the Melbourne Renegades at Adelaide Oval on Sunday afternoon. Tickets are now available.