Saints kicked seven successive goals on a run from early in the first quarter until halfway through the second term, to at one point lead by 39 on their way to a 11.12 (78) to 10.14 (74) victory.
GWS got within four in the last 23 seconds, kicking five of the six goals in the final quarter.
They left their run too late and the Saints ended a run of four successive losses to GWS, who dropped to 1-2 for the season after consecutive losses of their own.
It came after the start of the game was delayed by 50 minutes on Saturday due to lightning in the vicinity of the ground.
"I thought our group relaxed, they were singing Dancing in the Dark before they went out, they were pretty chilled," St Kilda coach Ross Lyon said.
"I think we deserved that win, we learnt a bit about how to time manage better in the end and there were some really courageous acts.
"We won't get ahead of ourselves, it's nice to get off the mark and pop that balloon."
The Saints superior pressure, better ball use and more accurate goalkicking had them well in front for most of the day, though they kicked just three goals after halftime.
"I think they dominated the first half I think we dominated that second half," GWS coach Adam Kingsley said.
Saints did dominate clearances and contested possessions early on, but GWS fought back in both areas.
St Kilda's forward half turnovers set up 39 of their 56 first half points.
"I'm not sure it was their pressure, I think we gifted them some goals, unfortunately through some poor decision making or poor skill," Kingsley said.
He lamented their inability to kick more than 2.7 in the third quarter, during which they had 15 more inside 50s and finished the term 27 behind, having scored just 5.13.
Jesse Hogan kicked the opening goal, but the visitors dominated the rest of the first quarter, slamming on five unanswered majors, two of them to Jack Higgins.
Saints kicked the first two goals of the second quarter before GWS produced some resistance.
Goals to Harry Rowston and Ryan Angwin briefly stopped the bleeding but Max Hall pounced on another GWS error to ensure the Saints won the second quarter and were ahead by 32 at halftime.
Jack Sinclair had 32 disposals, Nasiah Wanganeen-Milera finished with 27 possessions, five clearances and four score assists for St Kilda, who benefited from two goals each from Higgins and Alix Tauru.
Lyon said Anthony Caminiti suffered hamstring tightness.
Lachie Ash gathered 38 possessions and Clayton Oliver had 33 for GWS, with hogan and Phoenix Gothard kicking two goals each.
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