The Cats led at every change on Sunday before the Swans mounted a serious fourth-quarter challenge to reduce a game-high 21-point lead.
Lexi Hamilton brought the Swans within four points late in the last quarter, but the Cats held on to win 5.14 (44) to 6.5 (41) in front of 3,127 fans at Sydney's Henson Park.
Their second victory of the year comes past the halfway mark of the 11-round season, having started their campaign with one win, four losses and a draw.
Starting the round in 16th, Dan Lowther's charges left themselves with a chance of finishing as high as 11th by the end of the weekend.
Georgie Prespakis (19 disposals, 13 contested possessions) starred in the absence of injured ball winner Amy McDonald, alongside Julia Crocket-Grills (one goal) and Jacqueline Parry (one).
"It was hard to watch, but really exciting footy to watch," Lowther said.
"We've been in games up to our necks but we just haven't got across the line so the aim for today was to win.
"We move on to Richmond next week and (the aim) is the same. Whatever happens off the back of that (in terms of finals) is out of our control."
Sydney's season hangs by threads after their third consecutive loss.
Their path towards a second straight finals appearance won't grow any easier, with matches against competition heavyweights Brisbane and North Melbourne ahead.
Coach Scott Gowans lamented Sydney's inability to convert late after his side lost to Essendon earlier this week by the same margin.
They also went down to the Western Bulldogs by seven points in round five after staging another second-half challenge.
"It seems to be the story, doesn't it?" Gowans said.
"It's frustrating at the same time, but also you've got to be proud of effort and intensity, and if you bring that each week, you just never know.
"We won't look too much at where we sit on the ladder or anything like that."
Geelong brought the heat early against the Swans, kicking the opening three goals to build a 3.2 to 0.2 lead at the first change.
With the breeze behind them in the second term, Sydney finally found reward on the scoreboard when Irish import Paris McCarthy kicked three goals in seven minutes.
McCarthy stunned the Cats' defenders when she snapped overhead after teammate Bec Privitelli's shot at goal fell short to reduce the margin to two points.
The former Gaelic footballer, playing her first match of the season, had never kicked an AFLW goal before Sunday.
Ahead by just seven points, Geelong kicked the only goal in the third term before the Swans came alive in the last quarter.
Laura Gardiner (30 disposals, five clearances) was the star in a Swans engine room missing Ally Morphett and Chloe Molloy, alongside Sofia Hurley (20, two).
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