Chasing 131 on a tricky Gabba wicket on Friday, the Heat were teetering at 3/23 and looked set for another batting meltdown after losing trump cards Chris Lynn and Marnus Labuschagne in the first two overs.
But Joe Denly (41 off 40) was joined by the rock-solid Peirson and the pair all-but burst the Strikers' bubble in a 69-run stand that steadied the ship.
Peirson (47 off 44) and Joe Burns (17 off 17), who was dropped twice including from the first ball he faced, saw them home with six wickets and seven balls remaining.
Brisbane's victory was the second of four-straight needed in the finals series if they are to clinch a second BBL title, their next assignment a knockout against the Sydney Thunder in Canberra on Sunday.
Earlier the Strikers were stuck in their tracks after electing to bat, reaching just 1/53 after 10 overs.
Fresh off blasting 101 off just 62 balls a week ago, Adelaide star Alex Carey was in slow motion at the top of the order.
He had just nine off 19 balls at one stage, eventually falling for 13 off 22.
Their slow start forced a risky X-factor substitute from coach Jason Gillespie, who brought in batsman Jonathan Wells at the expense of unused strike bowler Daniel Worrall.
Things didn't improve though, Jake Weatherald (32 off 28) run out two balls after the break and captain Travis Head (12 off 21) also struggling.
Leg-spinners Mitchell Swepson (2/23 from four overs) and Labuschagne (3/13 off three) were again effective.
Labuschagne has taken nine wickets in four BBL games since returning from Test duty and has Ben Laughlin to thank for his most recent.
The veteran bowler sprinted 25 metres around the boundary before launching himself to pluck the ball one-handed while horizontal in the air to claim one of the catches of the tournament.
Adelaide needed something similar in the field but couldn't take a string of half chances after getting an early sniff when Lynn was knocked over by a beautiful Michael Neser delivery and Labuschagne followed in the next over.