After starting strongly on Wednesday night, the Demons were challenged stiffly but withstood everything thrown their way and were never headed in their 2.4 (16) to 1.8 (14) victory.
The Crows were left to rue a flat start and chronic inaccuracy, which ultimately sabotaged their bid to conjure a comeback triumph.
Georgia Campbell and Alyssa Bannan booted Melbourne's only goals of the evening, both in the opening quarter.
On-baller Tyla Hanks and winger Eliza McNamara, making a welcome return to the Dees' line-up, starred for the visitors, well supported by Sinead Goldrick and captain Kate Hore.
"We haven't had too many wins over here in nine seasons, so really pleased with the group for how we started the game," Melbourne coach Mick Stinear said.
"Super proud of the group.
"They've always been a tough team to play against, so this win will mean a lot to our group."
The Dees are on fire early 👀— AFL Women's (@aflwomens) #AFLWCrowsDees pic.twitter.com/onTwA7VteZOctober 9, 2024
Ebony Marinoff (29 disposals, 17 tackles) and Anne Hatchard (28 disposals) typically led the rearguard for the hosts, who were off-colour out of the blocks.
Early doors, Melbourne tackled aggressively, defended smartly and chopped Adelaide up repeatedly on the outside through slick handball chains.
The fifth-ranked Crows were also guilty of ill-discipline, a 50m penalty gifting Campbell the first goal of the evening, then a free kick for a push in the back resulting in Bannan's skilful goal from the scoreboard pocket.
Spearheaded by their dynamic duo Hatchard and Marinoff, the Crows improved noticeably in the second term.
They upped their tackling intensity and began dominating territory but were unable to capitalise with scoreboard pressure, conjuring 0.4 for the term, to only trim the Dees' advantage to nine points at halftime.
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Adelaide finally broke through for their first major, five minutes into the third stanza, when Madison Newman marked and goaled, helping reduce the deficit to four points at quarter time.
The Crows started the fourth period with a flurry of inside-50s which forced Melbourne onto the ropes.
But Adelaide couldn't land the killer blow.
Caitlin Gould missed a snap, the Demons backline rushed a behind and Chelsea Randall was off-line on the run, the three behinds edging Adelaide to within one point, before the visitors added one more behind through Ryleigh Wotherspoon to hold on grimly to the finish line.
"The effort was there but our execution was pretty shabby, really, across the whole game," Adelaide coach Matthew Clarke said.
"Defensively we were pretty sound - we held them to a couple of goals - but when you only kick one yourself, you're not going to win too many games."
Both teams will be in action again quickly, the Crows taking on GWS in Sydney and the Demons tackling St Kilda at Moorabbin, both on Sunday.
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