Garth took her four wickets at the expense of 23 runs off 20 deliveries.
Rockets were also indebted to Sophia Dunkley, who hit a third straight half-century at Trent Bridge.
She reached her 50 off 26 balls and when she was caught on the boundary by Davina Perrin off Alice Capsey for 65 off 37, Rockets needed 21 with 27 deliveries left for victory.
Phoenix, who lost the toss and were put into bat, mustered a below-par 7-122, losing wickets at regular intervals.
Phoenix never recovered from a shaky start as they lost Perrin, Tammy Beaumont, Capsey and Aussie captain Ellyse Perry (one) with 55 runs on the board. Emma Lamb top-scored for the visitors with a 34-ball 45.
In reply, Rockets lost Aussie Beth Mooney (seven) with the score on 20 before Dunkley, who struck two sixes and seven fours, put on 62 in 55 balls in partnership with Nat Sciver-Brunt. Sophie Luff (19 not out) saw them home.
Elsewhere, Orla Prendergast's new ball burst set Welsh Fire on course for a three-wicket victory over Manchester Super Giants.
There was also a key contribution from Aussie Heather Graham, who added 16 late on to avoid any further dramatics and add to her earlier haul of two wickets.
Ireland international Prendergast took three wickets in her first 10 deliveries at Old Trafford, with the hosts never recovering as they scraped together 9-85 in an under-powered first innings.
Fire also struggled for fluency, losing seven wickets in the chase, but got there, in part thanks to Graham, with nine balls remaining. I
Prendergast bowled Super Giant's Aussie captain Meg Lanning for a golden duck before adding Mady Villiers and Kathryn Bryce as she delivered decisive back-to-back sets. She later returned to see off India's Richa Ghosh to wrap up figures of 4-18.
Smriti Mandhana was run out after a miscommunication with another Aussie, Maitlan Brown, leaving Ryana MacDonald-Gay to top score from No.9 with 18 not out.
That remained the best individual effort of the match, but an opening stand of 27 from Australian Georgia Voll, caught by Lanning after reaching 16, and Sarah Bryce provided a sturdy enough platform to calm the visitors.
Lanning held two further catches, including one that dismissed top-scoring compatriot Georgia Wareham for 17, and spinner Sophie Ecclestone took 3-14 in a losing cause.
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