The Saints banked a third consecutive win with a gutsy 13.11 (89) to 10.12 (72) triumph at Adelaide Oval.
After St Kilda led by 31 points late in the second term, the Power surged to within five points with eight minutes remaining in the game.
But the spirited Saints responded with the next two majors - a classy long-bomb from the influential Jack Sinclair followed by a Lance Collard conversion after a strong mark.
St Kilda, powered by six consecutive goals in a first-quarter spree, are now entrenched in the top eight with three wins and a loss.
But the Power have slipped to a precarious one win and three defeats ahead of a looming Gather Round clash against a white-hot Hawthorn.
Saints forward Higgins booted two of his goals in the first-term blitz and teammate Mitch Owens kicked two for the match.
Veterans Sinclair (25 disposals, one goal), recruit Jack Macrae (28 possessions) and classy halfback Nasiah Wanganeen-Milera (25 touches) were instrumental in the victory.
Port's tall forwards Ollie Lord and Mitch Georgiades kicked three goals each, as did Jason Horne-Francis, who also collected 28 possessions in a dual midfield-forward role.
Power vice-captain Zak Butters gathered 30 touches in his first game of the season since recovering from a knee injury and defender Miles Bergman (25 disposals) was busy.
But the home side paid a huge price for their first-term lapse when unable to halt the Saints' six-goal burst.
Port's Horne-Francis literally walked into an open goal for the initial major of the game after a 50m penalty against Saint Marcus Windhager for entering the protected zone.
Power star Butters gave Windhager a serve which triggered a massive melee.
Butters - already the league's most fined player - will likely cop another financial hit and so could about 20 other players.
The fracas sparked the Saints, who kicked the next six goals in a 23-minute purple patch to create a 30-point lead.
The visitors led 6.4 to 2.4 at quarter-time but the Power made inroads by booting the initial two goals in the second term to trail by 13 points.
But the Saints kicked away again with three consecutive goals inside four minutes - Wanganeen-Milera goaled on the run from long-range to put his side 31 points clear.
St Kilda led by 23 points at halftime, 9.6 to 5.7, but the home side rallied with three successive majors - when Lord goaled after a free kick, they were 11 points down.
But Higgins converted a classy left-foot snap late to give his team a 17-point buffer at three-quarter time.
Port then charged with Lord and Georgiades scoring goals to trail by just five points and hold all momentum.
But Sinclair's brilliant goal on the run five minutes later steadied the Saints, who meet GWS next Sunday at Norwood Oval in Adelaide.