Phoenix captain Perry managed just five runs before Gardner made the catch off the bowling of Alexa Stonehouse.
Phoenix tallied 6-112, not enough to prevent Rockets pulling off a vital victory with three wickets and three balls to spare, though Perry did repay the compliment by running out Gardner during the visitors' innings.
The result meant Rockets drew level on points with third-placed Northern Superchargers to keep themselves in the hunt for a place in the knockout stage.
Birmingham, who started level with Trent in the table, won the toss and chose to bat, but lost Perry, Fran Wilson and Amy Jones in the opening 15 balls.
Phoenix's recovery was as impressive as it was crucial, with Sterre Kalis and India international Richa Ghosh putting on 95 for the women's competition's record fourth-wicket partnership.
Ghosh made 41 from 36 and Kalis 47 from 44 as they took the hosts to 6-112. Sciver-Brunt finished with figures of 2-16 and Australia spinner Gardner also picked up crucial late wickets to take 2-17.
Phoenix needed early wickets to help them defend their below-par total and they managed just that, with Bryony Smith and Nat Wraith dismissed in the first 20 balls.
That brought Sciver-Brunt, the leading run-scorer in The Hundred this year, to the crease,, and with Gardner also batting the visitors looked to be cantering to victory.
However, a flurry of wickets - including a timely run out of Gardner by Perry - made things suddenly appear much trickier for the Rockets.
The visitors perhaps benefited from a debatable no-ball call, which saw Josie Groves reprieved having initially been given out, but on this occasion they had enough to get over the line.
Katie George was there at the end alongside Groves to keep alive Rockets' hopes of going further in The Hundred.
