Carlton now face Collingwood, Hawthorn, West Coast and St Kilda.
The Blues, already without injured duo Tom De Koning and Blake Acres, suffered a blow before the opening bounce when key forward Harry McKay withdrew with illness.
McKay had been a focus during the week, having been cleared of concussion after a head knock against North Melbourne, with the Blues reprimanded by the AFL for their delay in taking him off the ground.
His late withdrawal on Friday left Curnow as Carlton’s lone key forward, and the superstar did his best to drag his team over the line.
Curnow kicked three goals in the first half as he threatened to break the game open but, like many of his team-mates, his influenced waned in the second half.
The puzzling call of the night was the decision to pull tagger Alex Cincotta off Zak Butters who he had blanketed for the first half. It allowed the star Port midfielder to work his way back into the game.
On the flip-side Willem Drew’s close checking of Patrick Cripps, and ability to win the ball himself, was critical as his side gathered momentum.
Ken Hinkley’s charges were almost left to rue a failure to seize their opportunity but fought back in style and held Carlton to just one goal in the second half.
Port Adelaide now sit behind the second-placed Blues on percentage alone after this boost to their top-four hopes with the chasing pack of Brisbane, Fremantle, GWS and Geelong.
Ollie Wines (26 disposals and 10 clearances) and Connor Rozee (24 disposals) were busy while Butters and Jason Horne-Francis helped lift their side late.
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Sam Walsh (23 touches) and Elijah Hollands (20 disposals) were prolific for the Blues while Jacob Weitering kept Charlie Dixon quiet.
Curnow drew first blood but Carlton otherwise spurned some good early chances, while luckless midfielder David Cuningham was forced out of the game with a shoulder injury.
Esava Ratugolea and Mitch Georgiades struck to give Port Adelaide a four-point lead at quarter-time.
But after the break, the Blues wrenched the game in their favour, finding multiple goalkickers to deliver a 25-point lead at half-time.
Port huffed and puffed in the third term, snagging two late goals to reduce the deficit to 10 points at the final change.
In the final term, Ratugolea and then Jackson Mead struck to give the visitors a slender lead.
Horne-Francis goaled from just inside 50 to all but ice the game, before a late Darcy Byrne-Jones toe-poke sealed the deal.
Port next host Sydney while Carlton play Collingwood.