Root and skipper Phil Salt put on 77 for the first wicket before Salt fell for 46, but Root was there at the end on 58 not out as his side reached their target of 138 for the loss of just three wickets with four balls to spare at Edgbaston.
Fire were indebted to Jordan Cox for two stunning relay catches on the boundary to dismiss Tasmanian Mitch Owen (18 off 14) and then Joe Clarke (35 from 23) as they reduced Phoenix from 1-53 to 5-79.
The hosts, who had elected to bat, did themselves few favours as Laurie Evans and Rehan Ahmed succumbed to disastrous run-outs.
They eventually managed to reach 9-137 with Clarke's 23-ball 35, which included two sixes and four fours, the most notable contribution as Lockie Ferguson and Marco Jansen took two wickets each.
Later on Saturday (Sunday AEST), Jofra Archer blasted through London Spirit's batting line-up to set up a five-wicket victory for Southern Brave at Lord's.
The England paceman took 4-9 runs to help reduce the hosts to 105 all out in just 92 balls.
Opener Jonny Bairstow was the only batter to offer any real resistance with a 43-ball knock 56, which included two sixes and five fours, with veteran Marcus Stoinis helping himself to 3-17.
Fellow Aussies Andrew Tye and Adam Zampa both fell to first-ball ducks, with the leggie one of Stoinis' victims.
Brave opener Jamie Smith looked to have put his side on the road to a comfortable win with 36 from 22 balls, but they collapsed from 2-55 to 5-60 in the space of 14 deliveries to give Spirit hope.
However, David Miller scored an unbeaten 39 and completed the chase with a second six in five balls with three still remaining.
With AAP.
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