The visitors won the toss and chose to bat, making a steady start, with Sophie Devine and Sterre Kalis going at a run a ball for an opening partnership of 32.
The introduction of Alana King, another Australian, saw the first breakthrough for the Rockets, trapping Devine lbw for 17.
That brought Phoenix captain Perry to the crease and she quickly upped the run rate alongside Kalis.
Perry raced to 41 from her first 20 balls and was making the most of the conditions, while also riding some luck in being dropped on the boundary in what ended up being an impressive 66 from 34 balls.
Her partnership with Kalis put on 102 from 60 deliveries - including 11 fours and four sixes - with every maximum coming from the bat of Perry.
Amy Jones arrived at the crease with four balls to go and Phoenix scrambled five more runs to leave their innings at 139 from their 100 balls, with Jones run out on the final delivery.
That left Kalis on 48 not out, with Birmingham setting a competitive total.
Perry remained in the action in the field, firstly taking the wicket of Grace Scrivens (6) and then catching Bryony Smith with a fantastic running effort at long on, with Rockets at 2-58.
Nat Sciver-Brunt came to the wicket in fine form and looked composed from the outset, reaching her sixth half-century in The Hundred with a powerful lofted drive off Hannah Baker from just 33 balls.
Trent Rockets looked well in contention at that point but the demise of Australian Ash Gardner and then Sciver-Brunt three balls later dragged Phoenix back into the game.
Rockets needed 27 from the last 10 balls but the game was not done yet.
Another Australian, Heather Graham, hit four boundaries, including a huge six off the 98th ball, to leave Rockets needing seven from the last two balls.
Natasha Wraith was run out on the penultimate delivery going for a second, leaving Graham needing a six to win from the final ball.
To the delight of the visitors, Graham could only manage two, taking her to a valiant 30 from 12 balls but confirming Birmingham Phoenix's first win in 13 attempts.
