The Dogs' Harvey Gallagher scored the go-ahead behind with only 24 seconds left in a tense 10.14 (74) to 10.12 (72) triumph at Adelaide Oval.
The Bulldogs (five wins, four losses) claimed eighth spot ahead of weekend games, while Port remain 13th with three wins and six defeats.
The Bulldogs were superbly served by on-ballers Ryley Sanders (31 disposals), skipper Marcus Bontempelli (30, one goal, 10 tackles) and Ed Richards (27 touches, one goal).
Dogs attackers Aaron Naughton, Rhylee West and Will Lewis booted two goals each.
Port's acting captain Zak Butters - against the club considered the frontrunner for his signature should he leave the Power at season-end - was typically prolific with 30 possessions.
Power spearhead Mitch Georgiades kicked 4.4, while halfback Kane Farrell (27 touches) and winger Jase Burgoyne (21) were busy.Â
The Bulldogs began brightly, scoring the initial three goals to race to a 20-0 advantage as the Power lost influential defender Miles Bergman to a toe injury.
Field umpire Daniel Johanson was also a casualty, felled by contact from Bulldog Ed Richards in the second minute and taken off the ground.
The Power's Georgiades booted two goals, both from technical holding free kicks against Buku Khamis, as the home side crept within seven points.
The Dogs increased their lead to 13 points, 6.2 to 4.1, at quarter-time via a late Naughton goal.
Port's Ewan MacKinlay reduced the deficit with an eye-catching major early in the second stanza, and when young teammate Jack Whitlock converted a 45m shot from a tight angle, the Power were only five points down.
The Bulldogs steadied with two strikes from Will Lewis to hold a handy 18-point buffer at halftime, 9.5 to 6.5, when the visitors led the inside-50 count 31-18 and Port's Bergman, after trying to return to action, was ruled out at the break.
The Power again made inroads in the third quarter, with Georgiades slotting his third goal and Logan Evans sinking a 55m bomb to reduce the margin to six points.
But after Evans' long-range launch in the 11th minute, both clubs couldn't score another goal - the Dogs recorded just one behind for the term and led by five points at three quarter time, 9.6 to 8.7.
The visitors gained breathing space six minutes into the final term when West ran into an open goal for a 13-point break.
Georgiades responded with his fourth goal about 10 minutes later and teammate Corey Durdin threaded a set shot to leave the hosts one point in arrears with two minutes and 34 seconds remaining.
The Power drew level with a rushed behind 30 seconds later, but the Dogs pressed forward in the dying stages, with Gallagher's checkside behind putting them in front, before Lewis scored another behind after the siren.
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