The Crows had looked poised to condemn the hosts to an eighth-straight defeat after building an 11-point lead before halftime at Ikon Park.
But the Tigers refused to relent and kicked six to two goals in the second half to set up a hard-earned 8.10 (58) to 6.11 (47) win.
Richmond had entered Sunday's round eight clash dead last in 18th (0-7) as the only team not to have won a game.
Ryan Ferguson's Tigers now move to 17th, ahead of Gold Coast, with their first defeat of Adelaide.
Meanwhile, the Crows' three-game winning streak is over and the three-time premiership side drop to fifth (5-2).
Conti (one goal, 35 disposals, six clearances) was outstanding in a thrilling midfield battle against Crows counterpart Ebony Marinoff.
Marinoff, who marked an equal-record 100 games last week, finished with 27 disposals and six clearances.
Ellie McKenzie was also busy for Richmond and kicked a career-best three goals from 24 touches, supported by captain Katie Brennan (two goals).
The Tigers ran hot from the first bounce and got off to a dream start when McKenzie launched near the arc for the opener.
But the Crows responded with the last two goals to take a five-point lead at quarter-time.
Key defender Gabby Seymour was substituted out in the second quarter with a knee injury, only adding more pressure on the Tigers' backline.
Danielle Ponter and Eloise Jones got on the scoreboard to give the Crows breathing space before halftime.
But the Tigers remained dogged and upped the pressure, punishing Adelaide by scoring all three goals in the third quarter from turnovers.
The Crows dominated territory in the final term but couldn't take advantage of their chances, kicking five behinds as Richmond prevailed.
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