The Australia star took the key wickets of Smriti Mandhana, Maia Bouchier for a duck, and Brave captain Georgia Adams in her first 10 deliveries.
Amy Jones, named player of the match, said: "Pez's (Perry's) start was unbelievable. When you have someone like that, who can deliver such performances, it is a huge lift for the team."
The Phoenix had lost 12 completed games in a row prior to Wednesday's victory over Trent Rockets but have now won two on the spin.
England wicketkeeper Jones snaffled six catches with the gloves - the most dismissals of any wicketkeeper in the history of the women's tournament - having earlier hit her maiden half century in the competition.
Starting the day filling the bottom two places in the table, both sides knew a win was essential to boost their qualification chances.
Phoenix were put in by Adams and a steady start was only interrupted when 16-year-old slow-left armer Tilly Corteen-Coleman struck with her first ball to have Sophie Devine trapped lbw.
Corteen-Coleman struck again as Rhianna Southby effected a smart stumping to get rid of Sterre Kalis, forcing Perry and Jones to consolidate.
They took the total past 50 before Perry tried to beat long-on but only found the safe hands of Bouchier off Adams.
Jones accelerated with 51 from 36 balls, aided by a cameo of 22 in 12 deliveries from Emily Arlott that included two sixes.
Jones' half-century saw Phoenix reach 7-137, with Lauren Cheatle, another Australian, taking 1-14 from her 20-ball allocation.
After a three-boundary blitz from the returning Mandhana at the start of the Brave innings, Perry, the Phoenix captain, took over to finish with 3-21.
Meanwhile, Australia's Jess Jonassen deployed her ever-accurate left-arm spin to take 2-29 to help Welsh Fire to a six-wicket win over Trent Rockets at Nottingham.
Jonassen also weighed in with 26 in Fire's reply to a Trent target of 5-130. The Rockets had been helped by contributions from Australia's Ash Gardner (31) and Heather Graham's (27 not out).
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