Adelaide, the great escape artist, came from behind to pip Melbourne by 17 points and climb to fifth on the ladder.
The Crows, after trailing for most of the thrilling match at Adelaide Oval on Saturday afternoon, booted the last two goals for a precious 11.13 (79) to 9.8 (62) victory.
With a fourth win by 17 points or less this season, Adelaide stole fifth spot from the Demons, who slip to sixth.
Melbourne led at the quarter breaks by eight, one and two points, but key Crow Riley Thilthorpe kicked the go-ahead goal 22 minutes into the final term.
And Josh Rachele slotted his third major on the final siren to steer the Crows to a 9-5 win-loss record.
Adelaide skipper Jordan Dawson (18 disposals) was relatively quiet but kicked two goals, Rachele's triple treats came from 17 touches, and Darcy Fogarty booted two goals.
Crows defenders Wayne Milera (27 disposals, one goal) and Rory Laird (26) were outstanding, and Sam Berry's breakout season continued with 24 touches, seven clearances and eight score involvements.
Melbourne Captain Max Gawn was superb with 28 disposals and 31 hit-outs, and Jacob van Rooyen kicked three goals in a first-term blitz.
And while Kysaiah Pickett was restricted to just 15 disposals by James Peatling and forced into attack in the frenetic final term, Jake Lever, in his 200th AFL game and against his former club, was a standout in defence.
His teammates Jake Bowey (24 touches) and Daniel Turner (22) were other chief ball-winners for the visitors, who have a bye next weekend, when the Crows face home-town rival Port Adelaide.
The Demons, sparked by van Rooyen, gained an early edge by kicking 5.0 to 3.4 in the opening term when Adelaide's Izak Rankine, after a two-game injury absence, set up two of the Crows' majors with class passes.
Adelaide kicked the only two goals of a tight second stanza, with Dawson's conversion on the run from 55 metres a highlight.
Melbourne's first-term accuracy deserted it, scoring 0.6 for the quarter and it led at half-time 5.6 to 5.5.
Each team kicked three goals in a proverbial arm wrestle of a third quarter when the Crows took the lead briefly.
But double strikes from Melbourne's Kade Chandler gave the visitor a two-point buffer at three-quarter time, 8.8 to 8.6.
Goalsneak Rachele then took centre stage with two goals in the last quarter, punctuated by Thilthorpe's major, as the home side banked a hard-fought triumph.
SCOREBOARD
ADELAIDE 3.4 5.5 8.6 11.13 (79)
MELBOURNE 5.0 5.6 8.8 9.8 (62)
GOALS
Adelaide: Rachele 3, Fogarty 2, Dawson 2, Walker, Thilthorpe, Milera, Keays
Melbourne: van Rooyen 3, Langford 2, Jefferson 2, Chandler 2
BEST
Adelaide: Keane, Berry, Dawson, Borlase, Milera, Rachele
Melbourne: Gawn, Chandler, Sparrow, van Rooyen, Turner
