Phoenix won the toss and chose to bowl but Perry's fellow Australia international Georgia Voll knocked 78 from 55 balls to put the Fire in a commanding position while Sophie Devine's 35 from 19 balls helped propel them to 3-144.
In reply, Davina Perrin and Tammy Beaumont were dismissed early which brought Capsey and Perry to to the crease and it looked like the pair would take Phoenix to an easy victory until Capsey was caught by Aussie Heather Graham off the bowling of Voll.
Phoenix still needed 24 for victory from 18 but the Fire accumulated a flurry of dot balls which left Birmingham needing 10 from the final three balls.
However, Perry was still at the crease and she slogged her international team-mate Graham over the boundary at deep midwicket before she cracked the next ball through point to give Birmingham a dramatic first win with a single ball to spare.
Meanwhile, Australia skipper Sophie Molineux and England opener Maia Bouchier teamed up to condemn London Spirit to a heavy defeat.
Molineux took three wickets for just seven runs as Southern Brave limited their opponents to 8-114 Â at Lord's before Bouchier's 46 saw them reach their victory target with five wickets and seven balls remaining.
The Victorian removed Amy Jones, Marie Kelly and Deandra Dottin with Tilly Corteen-Coleman and Sarah Glenn taking two wickets each in a dominant bowling performance despite South Africans Marizanne Kapp hitting 34 and Nadine de Klerk 38, including three maximums.
Bouchier anchored Brave's reply, falling four runs short of a half-century when she was stumped off the bowling of Charlie Dean with just one run required.
Captain Molineux departed for 23 to the very next ball and it was left to Naomi Dattani to finally nudge her side across the finishing line.
