Laird (28 possessions) was superb in defence and forwards Tom Lynch, Darcy Fogarty and Riley Knight each booted two goals.
Port kicked four of the initial five goals but then a slick Adelaide outfit seized control in 40-degree conditions.
The Crows led by 36 points midway through the last quarter, but the Power rallied with five unanswered goals to sneak within five points before Adelaide kicked the last two goals of the game.
The Power’s reigning club champion Justin Westhoff, Steven Motlop, Brad Ebert and their top pick at last year’s draft, Connor Rozee, kicked two goals each.
Another of Port’s highly rated draftees, Zak Butters, was a standout with 25 disposals and fellow newcomer Xavier Duursma kicked a goal.
Defender Dan Houston gained 21 disposals and Travis Boak, who has stood down as captain this year, was busy with 27 touches.
Port’s new signing Scott Lycett, who arrived after winning last year’s premiership with West Coast, had the 24 hit-outs – the most of any player – while former Hawthorn halfback Ryan Burton collected 17 disposals.