The Crows trailed by 13 points early in the second quarter but kicked the final nine goals of the game, with Randall jagging three, to claim a thumping 10.5 (65) to 3.9 (27) win at Unley Oval.
Randall missed the Crows' season-opening loss to St Kilda as she was the celebrant at her brother's wedding in Bali.
She got Adelaide going when she burst away late in the third term to cut the three-quarter time deficit to just three points.
But the real sliding doors moment arrived early in the final term.
Geelong star Aishling Moloney could have kicked a goal or handed the ball off to extend Geelong's lead to nine, but was instead run down by Grace Kelly.
The Crows took it down the other end and Eloise Jones put Adelaide in front with the first of seven unanswered goals for the quarter.
Adelaide never looked back as they levelled their ledger at 1-1, while condemning the Cats to an 0-2 start.
"We started a little bit slower there than what we would have liked to," Randall told Fox Footy.
"But we started to get our contest and pressure back at a level that we were really happy with, and then just some of our execution.
"In all honesty, I think we finally found a little bit more groove, which was really pleasing to see. That was definitely the level."
Randall (three goals), Jones (two), Caitlin Gould (two) and Danielle Ponter (two) hit the scoreboard while Ebony Marinoff (30 disposals, 10 tackels) and Niamh Kelly (434 metres gained from 17 disposals) were prolific.
Georgie Prespakis (30 touches, nine clearances) and Rebecca Webster (27 disposals) were busy for Geelong.
Moloney, last year's AFLW leading goal kicker, nabbed two early majors, including a goal of the year contender, where she audaciously snapped the ball over her head.
But from there, Adelaide found their feet.
Adelaide next play GWS at Henson Park on Saturday, when Geelong host Sydney at GMHBA Stadium.
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