Crows captain Jordan Dawson won the Showdown medal as star recruit Izaak Rankine, young key forward Riley Thilthorpe and stalwarts Rory Sloane and Rory Laird shone in the 18.9 (117) to 13.8 (86) win on Saturday night at Adelaide Oval.
It is Adelaide's first win of the season and leaves the fierce local rivals with 1-2 records.
Rankine kicked a goal after the final siren to cap his star performance, with the former Gold Coast forward kicking four majors.
With Darcy Fogarty injured, Thilthorpe stood up and kicked a career-best five goals.
Port had won five of the past six Showdowns and also had a major victory ahead of this match, striking a deal with Collingwood that allowed them to wear their traditional black-and-white "prison bar" guernsey.
The Power started favourites in their home game, but the Crows ran over the top of them after a pulsating duel, with seven goals to two in the last term.
Adelaide sprung a tactical surprise, playing Dawson in the midfield rather than his usual defensive role, and he was massive early.
Dawson was mighty with 12 contested possessions, but Rankine, Laird and Thilthorpe had even bigger games in a famous Crows win.
Port kicked the first two goals, but Adelaide hit back immediately in a frantic start that set the stage for a titanic struggle.
There was controversy when Adelaide defender Lachie Sholl took a mark, went slightly off his line and was tackled by Sam Powell-Pepper, who kicked the goal when it should have been a Crows clearance out of danger.
After taking a one-goal lead into quarter time, Port broke out to a 17-point advantage in the second term and looked like they might be about to take control.
But the Crows kicked the last three goals of the half and, fittingly, scores were level at the main break.
The intensity of the match was summed up at the start of the third term when Crows teenager Max Michalanney was ironed out by Rowan Marshall.
Michalanney, Adelaide's first father-son recruit, was charging at full pace after the ball when Marshall came from the other direction and cleaned him up with a brutal front-on bump.
It was a fair hit that sent Michalanney to the bench with sore ribs, but the 19-year-old sucked it up and returned to the field to be among the Crows' best.
With the pressure enormous, the two teams went goal-for-goal before Adelaide blew some chances late in the third term and led by three points at the last change.
Port kicked two of the first three goals in the last quarter and led by four points at 10 minutes.
Junior Rioli dropped a mark in attack and Rankine then won a critical ball on the wing, setting up Lachie Murphy for a goal that gave Adelaide back the lead.
That opened the floodgates, with the Crows kicking the last six goals of the match.
Conor Rozee starred for the Power as Powell-Pepper and Charlie Dixon kicked three goals apiece.
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