Cook's stunner on the run delivered the Crows a stirring 11.10 (76) to 11.9 (75) victory at Adelaide Oval on Friday night.
Cook enters folklore in the closest-ever Showdown with a 45m shot to pinch victory.
Just 35 seconds earlier, young Power forward Joe Berry snapped a left-footer to give his side a five-point lead.
But Cook's late heroics lifted the Crows to a famous win and a 4-4 win-loss ledger this season.
"There's something special about these games ... they just don't disappoint," Adelaide coach Matthew Nicks said.
Port slip into a 3-5 record after another classic fixture of the South Australian rivals full of late drama.
"It's hard to unpack," Power coach Josh Carr said.
"That's a great game for all of our players to gain some some great learnings from (but) obviously we're disappointed."
Adelaide halfback Wayne Milera continued his superb form with 29 disposals, Sam Berry (29 touches) laid 19 tackles, and Jake Soligo (22 touches) and Izak Rankine (23, one goal) were influences.
Ben Keays potted three goals and Josh Rachele and Toby Murray kicked two each, with Riley Thilthorpe (one goal) well held by Port's Aliir Aliir (14 marks, 21 disposals).
The Power's prime movers were Zak Butters (30 disposals), Ollie Wines (30) and Jason Horne-Francis (20), while Mitch Georgiades kicked three majors, and Corey Durdin and Berry two apiece.
Port, despite having five less inside 50s, held a 3.2 to 1.2 quarter-time lead with Georgiades among their goalscorers.
The Crows lost former captain Taylor Walker to a right hamstring injury but, after goals from Keays and Rachele, drew level with Port at 22 all.
Power acting skipper Butters restored a slender six-point break after nailing a 45m boundary line set shot from a dubious pushing free against Soligo.
Port held a 4.4 to 3.4 halftime buffer with Adelaide's brilliant halfback Milera (19 touches) and Butters (18) both outstanding.
The Power held a 13-point advantage midway through the third term before the Crows gained traction late.
After a stunner on the run from Rankine, his teammate Keays snapped his second before third-gamer Murray stole the spotlight.
The younger brother of Nick, who was dropped for the game, twice got on the end of Soligo passes and, twice converted clutch set shots.
The Murray retort helped Adelaide establish an eight-point lead at three quarter-time, 8.9 to 7.7, after booting five goals to three for the term.
The industrious Keays highlighted his impact with a third major early in the final term as the Crows went 15 points clear.
And Adelaide appeared home when Thilthorpe, after a shrewd back-handed tap from Rachele, curled a left-footer through for a 21-point break five minutes in.
But like so many Showdowns, there was a dramatic twist.
Georgiades slotted two of Port's next four consecutive goals - they crept within two points with about six minutes remaining before Berry's snap gave them a late five-point lead.
But Adelaide frantically pressed forward and Cook delivered the game-breaker in the final seconds.