Skipper Ash Dixon was then denied one of the best tries this season after the hooker broke loose from within his own half and put in a chip and chase.
But Bay of Plenty flanker Mitch Karpik just managed to ground the ball before Dixon could.
He did get on the score sheet in the 28th minute though, getting a more typical try off the back of a classic Magpies lineout drive.
Hawke's Bay were turning the screws with some sustained attacking phases but a loose pass was intercepted by Steamers winger Emoni Narawa who coasted 80 metres to score, completely against the run of play.
That meant the Magpies led by only two points at halftime despite having a major advantage in both territory and possession.
The second half started far more evenly, with Hawke's Bay first five Caleb Makene missing a chance to extend that lead with a penalty in the 52nd minute.
And the door opened for Bay of Plenty a couple minutes later when Jonah Lowe was yellow carded for taking Narawa out in the air.
It took the majority of Lowe's 10 minutes on the sideline, but the Steamers finally cracked the Magpies defence and took the lead with another try to Trask in the 65th minute.
The hosts took control of the game, but Trask couldn't extend the lead past a converted try with a penalty shot.
The Magpies laid siege to the Bay of Plenty line over the final five minutes through a series of lineout mauls.
Steamers prop Ross Geldenhuys was yellow carded for collapsing one of them, but the hosts were able to hold on for the win.
The Magpies are home at McLean Park again next week, with Wellington coming to town to challenge for the Ranfurly Shield.
Magpies coach Mark Ozich said his side's performance was disappointing.
"That's definitely one that got away," he said.
"We're our own worst enemies unfortunately, from my point of view our execution on the day, our skillset let us down a bit."
He said Emoni Narawa's intercept try before halftime proved crucial as the Magpies had been hot on attack at that point in the match.
Ozich was disappointed his team couldn't pick up a fourth try after having scored three before halftime, as Bay of Plenty held them to zero points after the break.
"It's definitely one that hurts because if we were just marginally better, we could have come away with five points [for a bonus point win] there but it wasn't to be."
He said Hawke's Bay's fundamentals and skill executions were just not good enough in the second half:
"It's very disappointing for me as a coach to have to watch that, and I imagine for people watching at home."
Ozich said a similar effort next week would not be enough to compete with a Wellington side coming to McLean Park to take the Ranfurly Shield.
"If we don't reflect as a group and as individuals, reflect on our preparation and our execution of what we're supposed to be doing, we'll be nowhere near it next week," he said.
In-form Magpies winger Lolagi Visinia went off with a head injury, so could be unavailable for that match depending on how he comes through concussion protocols.
There are also niggles for loose forwards Gareth Evans (ankle) and Devan Flanders (shoulder) that could keep them out as well.
Bay of Plenty 22 (Kaleb Trask 2, Emoni Narawa tries; Trask 2/3 con, 1/2 pen) Hawke's Bay 17 (Lolagi Visinia, Jonah Lowe, Ash Dixon tries; Caleb Makene 1/3 cons, 0/1 pen).
HT: 15-17
