Coming off their first loss of the season to reigning premiers Brisbane, Adelaide never led by more than seven points on Friday en route to a hard-earned 3.8 (26) to 3.4 (22) victory.
The win lifted the Crows, now 5-1 for the season, to within two points of ladder leaders North Melbourne.
St Kilda (3-3) suffered a third straight defeat, but cling on to eighth spot on percentage ahead of Essendon.
Adelaide maintained their perfect record against the Saints, notching a fifth straight win and ensured they didn't suffer successive losses for the first time since 2020.
The Crows tallied 14 more inside 50s and 21 extra contested possessions, but struggled to fashion scoring opportunities against a doughty Saints defence, which pulled off numerous smothers.
The Saints trailed after each quarter but also had problems in the forward half, taking just two marks inside 50.
They got within one point with just under nine minutes to go when Hannah Stuart goaled.
The Saints didn't manage another score, while the Crows added three more behinds to clinch victory.
"There was lots to like other than perhaps just our ball use and our ability, to make the most of our opportunities," Crows coach Matthew Clarke said.
Adelaide dynamo Ebony Marinoff racked up 31 touches, 17 tackles, 16 contested possessions, nine clearances and six marks.
The Saints' run of low scores continued, after they kicked just two goals in each of their last two matches and three in the one before that.
The Saints kicked the first goal of Friday night's match, with Molly McDonald putting through a set shot after taking a mark.
But they didn't kick their second until the last 25 seconds of the third quarter when J'Noemi Anderson registered a major.
That levelled the scores, but Chelsea Randall's last-second kick bounced over a couple of players to give the Crows a six-point lead heading into the last quarter.
Adelaide's Hannah Munyard kicked one of the best AFLW goals of the season, sprinting away just inside the boundary line in the first quarter and goaling with a checkside kick from an acute angle to give the Crows the lead.
Tyanna Smith worked hard for the Saints, notching 22 touches, 14 tackles and eight clearances.
"I was really proud of our group I thought we really stuck at it, it was a really hard fought game," Saints coach Nick Dal Santo said.
"We're pleased with some aspects of our game but we'd like to score more goals."
Adelaide's Abbie Ballard, who only recently returned from an ACL injury, was forced out of the game and was seen with an ice pack strapped to her left knee .
"She's got a history there, so we were ultra-conservative. So she will have to get a scan and we'll see where this lands," Clarke said.
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