Cox rounded off a good week after being selected for England's Test squad by scoring an unbeaten 61 off 30 balls with nine fours and two sixes, one of which sealed a comprehensive victory with 37 balls remaining.
The result was all but settled by the halfway mark, with the already-eliminated Spirit bowled out for just 96 after being sent in.
Michael Pepper started well for the visitors, attacking the powerplay for an 18-ball 27, but when he fell - bowled by Zampa - the die was cast.
Twenty-one balls later Spirit were floundering on 5-52, with Ollie Pope, Shimron Hetmyer and Andre Russell back in the hutch.
They never recovered, with Zampa continuing his fine tournament with 4-17 while Sam Curran (3-10) and Tom Curran (221) offered good support.
Olly Stone - also included in the England squad alongside Cox for the upcoming series against Sri Lanka - ran in hard as Spirit sought to defend 96, dismissing Will Jacks for two.
But it was a case of when and not if for Invincibles, and Cox and Dawid Malan (29 not out) got them over the line with room to spare.
The win means Sam Billings' side, who have won six of their seven games, are guaranteed a place in the top three of the men's competition.
Their final game against Trent Rockets will settle whether they will head straight to next Sunday's final or if they will need to progress through the eliminator the day before.
