At a damp Go Media Stadium on Saturday, South Australian Lachie Brook opened the scoring for the home team when he fired the ball into the bottom left corner.
But Auckland FC lost their spark after halftime, and United grabbed an equaliser after Crawford poked a superb cross from defender Yaya Dukuly past goalkeeper Michael Woud in the 63rd minute.
The Reds will aim to capitalise on their home advantage in the second leg in Adelaide next Friday, with the winner to progress to the grand final against either Newcastle Jets of Sydney FC.
Auckland won't just count the cost of surrendering their lead though, after Uruguayan midfielder Guillermo May landed awkwardly when he thrust a ball up the field and was left clutching his Achilles in the second half.
But their quest for a maiden championship remains intact.
"I've loved the club (Adelaide United) since I was a kid, so it was a bit of a weird moment (to score against them)," Brook told Paramount+.
"It's all to play for in the next game now."
For Adelaide, they were without their top scorer Luka Jovanovic, after he had conceded two yellow cards, including one for taking his shirt off when he netted the winning goal, against Melbourne City on April 26.
Jovanovic will return for the second leg, with Adelaide knowing that a spot in their first grand final in a decade is within their grasp.
"It would have been nice to get a win, but it was nice to get a goal to help the team out," Crawford said.
"It's in our hands. Hopefully, we can get a result and go to the grand final."
The Kiwis slowly built into the contest, and Brook capitalised on a loose ball in the box before thumping it past goalkeeper Joshua Smits in the 24th minute.
Auckland thought they had a second goal when May guided the ball into the back of the net from a Smits save, but he was ruled offside.
After halftime, the home side was forced into a reshuffle after May departed, before midfielder Cameron Howieson left moments later after copping a hand to the eye.
Adelaide equalised when Dukuly assisted Crawford, setting up a tense finish as both teams searched for a winner.