The Heat slumped to their third loss from three games this summer despite a tremendous effort by stand-in captain Jimmy Peirson who scored 69 off 36 deliveries but couldn't get the hosts to their target of 151.
Instead, the Strikers held on to defend their 6-150 total with the Heat finishing on 9-148 in reply, with Peirson thrust into the captaincy after talisman Chris Lynn missed the game following a hamstring injury in the warm-up.
Adelaide were perhaps a few runs short with the bat, but a remarkable performance with the ball saw them reduce Brisbane to 8-68 in reply, as Khan, Briggs, Wes Agar, Daniel Worrall and Peter Siddle all contributed strongly.
Peirson's quickfire knock took the fixture down to the wire, but needing 13 off the final over, the Heat batsman three times turned down a run and couldn't score a six off the last ball to take the match into a super-over.
The win, which included securing the Bash Boost Point, sees Adelaide leap into third position in the BBL10 standings.
The Strikers also suffered an injury blow with Afghan leg-spinner Rashid Khan (2-30) limping off late with an apparent leg complaint.
Adelaide will be hoping to have Australian wicketkeeper Alex Carey and quick Harry Conway back in their squad after the duo failed to beat border closures due to NSW's COVID outbreak and had to miss Wednesday's match.
The Strikers will now fly home to enjoy a short break before lighting up Adelaide Oval against the Perth Scorchers on 28 and 31 December. Tickets are available now, with the iconic venue set to be pumping as 2020 finally draws to a close.