The Bulls reeled in the target of 414 late on Monday - the highest fourth-innings chase at Bellerive Oval - with four wickets to spare.
Peirson (75no) put on an unbeaten 143-run stand with allrounder Michael Neser (76no) to guide the Bulls home.
It was a remarkable turnaround for the reigning Shield champions, who were routed for just 107 in their first innings.
"We had to dig deep to get it to day four, and then to get a win is the stuff fairytales are made of," Peirson said.
"We knew if we could bat long enough on day four, we'd have a chance."
The Bulls resumed day four at 2-164, still needing another 250 to win.
They looked on the ropes at 5-233 after Tigers quick Riley Meredith (3-70) took wickets either side of lunch.
But Peirson and Neser rode their luck in the final session as the Tigers shelled three catches.
Peirson was dropped by Jordan Silk at slip off Jackson Bird and received a second life when George Bailey grassed a difficult leaping chance at short cover.
Former Test batsmen Joe Burns (74) and Marnus Labuschagne (78) set up the win at the top of the order.
Tasmania posted 167 after being sent in on day one before reaching 353 in their second innings via a Bailey century.
Burns had the Bulls' chase off to a solid start on day three, after they lost Matthew Renshaw for an aggressive 27 off 24 balls.
Tigers coach Adam Griffith knew his side were in for a challenge on a pitch that had flattened out after 18 wickets fell on day one.
"It probably sums our season up so far," he said of the loss, admitting his side were about 60 runs short in their second innings.
"We had our opportunities and played some really good cricket but, when the chance was there to win the game, we couldn't quite grasp it."
Queensland's 6-414 in just over four sessions was the Shield's fifth-highest run chase.
The previous best fourth-innings chase at Bellerive Oval was in 1997-98, when Tasmania reached 3-401 needing 399 against Victoria.
The Bulls' second win of the season lifts them to third on the ladder behind Victoria and NSW going into the BBL break. The Tigers are in fifth place.