The Crows came from 13 points down in the third quarter to triumph 9.9 (63) to 7.12 (54) at a rain-soaked Adelaide Oval on Saturday night.
Adelaide (56 premiership points) are fifth on the ladder, just 0.2 per cent behind fourth-placed Brisbane. Sixth-placed Melbourne are also on 56 points.
"Relief is not the right word," Crows coach Matthew Nicks said.
"We just had a brief chat as a group ... if you play that way and win, maybe it's a good sign."
Star Crow Izak Rankine was a standout with a career-high 37 disposals, one goal and a dozen clearances, and skipper Jordan Dawson was influential with 35 touches and a goal.
Adelaide's Callum Ah Chee booted two vital goals in the third quarter, Taylor Walker also kicked two majors, and midfielder Sam Berry (24 touches) was prominent.
Richmond's Seth Campbell bagged three goals, teammate Mykelti Lefau scored two, Jack Ross (28 touches) and Patrick Retschko (24) were busy, and midseason rookie draftee Kye Annand impressed with a match-high eight marks.
The Tigers led 2.2 to 1.3 at quarter-time, with Adelaide's flat opening prompting coach Matthew Nicks to give his players a serve at the break.
The Crows scored three goals to none in the initial 14 minutes of the second term to create a 12-point advantage but, as heavy rain fell to make an error-riddled contest even more challenging, the Tigers hit back.
Steely Green and Campbell booted goals to shrink their deficit to one point at halftime.
Campbell's influence magnified in the third term when he slotted two more goals to give the Tigers a 13-point break and alarm the 42,762-strong pro-Adelaide crowd.
But the Crows responded with two moments of magic.
First, prized recruit Callum Ah Chee kicked a sensational goal on the run from 55 metres in the driving rain.
And 10 minutes later, the dynamic Rankine scored a classy goal from 50m out, also on the run.
Rankine's stunner was followed by another Ah Chee major as Adelaide reclaimed the lead - four points ahead at three quarter-time, 7.7 to 6.9.
Adelaide veteran Taylor Walker extended the home side's lead to 10 points before the Tigers rallied again.
A Hugo Ralphsmith goal was followed by Sam Lalor missing a close-range set shot that would have put the visitors in front.
Instead, they were four points down and Adelaide's Zac Taylor sealed victory with a 50m set shot with under four minutes remaining.
"We let ourselves down - execution, discipline, things we can control," Richmond coach Adem Yze said.
"I couldn't fault our players, our spirit, our grit, our fight ... but we've got to be really honest."
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