The Renegades had crawled to only 56 runs from their opening 12 overs until Cox kickstarted their innings with a huge six in the power surge on Thursday night.
The hard-hitting England batter was the main aggressor in a critical 66-run stand with Jonathan Wells for the fourth wicket, until lofting a quicker ball from Riley Meredith (1-34) to Chris Jordan at mid-on.
Wells (38 not out) put his foot down from there to steer the under-fire Renegades to a total they will hope to defend against a side that tends to struggle on the road.
The Hurricanes had the hosts on the ropes from the opening over when Tim David dismissed Shaun Marsh caught behind for a duck to complete a wicket maiden.
South Africa star Quinton de Kock (20) looked set to reignite the Gades' innings in his final BBL match this season, until giving up his wicket with a timid slice to deep point.
Rising star Jake Fraser-McGurk (26) was next to make a solid start, but after smacking a powerful boundary he was out next ball after picking out Jordan at long on.
Hobart left-arm spinner Patrick Dooley (0-16) was a key to restricting the Renegades' top order, delivering 15 dot balls in four tight overs.
David was not called on to bowl a second over, despite claiming one scalp without giving up a run to start the match.
The Gades entered the clash without captain Nic Maddinson, with Will Sutherland instead taking the reins.
Former Australian one-ball skipper Aaron Finch was also left out of the XI.
Finch revealed before the match that this season will be his last in the BBL, as the second-highest run-scorer in the history of the competition winds down his career.
The Hurricanes are four points behind the Sydney Sixers (third) and the Melbourne Stars (fourth), but had two matches in hand ahead of their clash with the Renegades, as they look to keep pace with the top four.Â
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