Five Crows booted multiple goals in a 13.11 (89) to 12.12 (84) victory before a bumper 53,117-strong crowd at Adelaide Oval on Saturday night.
Adelaide led by two points at three quarter-time and made a defining move with three goals in a four-minute burst in the last term - two in 60 seconds from Izak Rankine.
But just when the Power appeared out, Port booted two late goals - the last from Jeremy Finlayson reduced their deficit to four points with one minute 47 seconds remaining.
The Crows survived the frantic finale to edge the Power 29-28 in the overall record, but defender Mark Keane was reported.
The Irishman - who took a courageous contested mark to deny a Port attack in the last minute - pushed out with his legs, connecting with the midriff of Finlayson, in the third quarter.
Adelaide retain a grip on fifth spot with Riley Thilthorpe and Darcy Fogarty slotting three goals each and Rankine, Josh Rachele and Ben Keays booting two.
Thilthorpe was an immense aerial presence, grabbing 10 marks - half of them contested - while ex-Giant James Peatling (24 disposals) and defender Josh Worrell (23) were influential.
Port skipper Connor Rozee was superb at halfback with 32 touches and a goal and his deputy Zak Butters (33 possessions) was typically busy.
Miles Bergman's tag on Jordan Dawson (14 disposals) paid off, Finlayson slotted three goals and Mitch Georgiades, Darcy Byrne-Jones and Sam Powell-Pepper scored two apiece.
Adelaide opened with a dominant quarter-hour for a 15-1 lead but led by only three at quarter-time, 3.4 to 3.1.
An after-the-siren conversion from Crow Josh Rachele gave Adelaide a five-point buffer at halftime, 7.8 to 7.3.
The tide turned towards Port early in the third quarter with Rozee scoring after a 50m penalty against Rankine rapidly followed by a Powell-Pepper strike.
The Power led by nine points before the Crows rallied with the next three goals - the first from Thilthorpe after another strong mark - for an eight-point lead.
But a stunning snap from a tight angle by Jason Horne-Francis reduced Port's deficit to just two points at three quarter-time.
The Crows then kicked three goals in four minutes and were 21 21 points up before Port's late flourish - Byrne-Jones and Finlayson goaled only for their side to fall just short.
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