Play did not get underway until well into the Leeds Saturday evening but when the action started, Australian wickets fell quickly to raise England hopes of ensuring their target would be more than reachable.
Seam-bowling all-rounder Chris Woakes, playing his first Test since March last year, took 3-68 as Australia were dismissed for 224 in their second innings on a rain-marred third day.
That left England, who trail 2-0 in the best-of-five series, needing 251 to win, with more than two days to get the runs on a ground where there have only been five previous successful chases in excess of 250 to win a Test.
At stumps they were 27-0 off just five overs, with Crawley (nine not out) and Duckett (18 not out) having struck four fours between them.
Ashes holders Australia were earlier indebted to Travis Head's 77.
After a lengthy rain delay, the match resumed at 1545 GMT with Australia 116-4.
Grey skies above made for difficult batting conditions, although England were a bowler down, with Ollie Robinson not used following his first-innings back spasm.
Nevertheless, Mitchell Marsh -- who had marked his first Test since 2019 with a run-a-ball 118 in Australia's first innings -- drove a Stuart Broad inswinger down the ground before square-cutting Woakes for another four.
Woakes, however, had his revenge with a burst of two wickets for seven runs in 13 balls.
He first dismissed Marsh (28) when the all-rounder tried to withdraw his bat to an outswinger only to edge to wicketkeeper Jonny Bairstow.
New batsman Alex Carey walked out to a chorus of boos from a crowd still upset by his role in the controversial stumping of local hero Bairstow during Australia's 43-run win in the second Test at Lord's last week, when the Yorkshireman believed the ball was dead.
But the jeers turned to raucous cheers when Carey fell for five as he also tried to leave a rising delivery from Woakes only to glove the ball onto his stumps to leave the Aussies 139-6.
Mark Wood's outright speed had helped the injury-prone Durham fast bowler mark his return to Test cricket with 5-34 in Australia's first innings.
And he struck again when Mitchell Starc skyed the ball for Harry Brook to take a fine diving catch.
Wood then had Australia captain Pat Cummins, often a dangerous lower-order batsman, caught behind for just one after he fended outside off stump.
But new batsman Todd Murphy, in his first game of the series after fellow spinner Nathan Lyon's tour-ending calf injury at Lord's, stylishly cover-drove Wood for a first-ball four.
Head then pulled and carved Woakes for boundaries before smashing Wood through midwicket to complete a valuable fifty with his sixth four in 94 balls.
The left-hander, whose stand of 155 with Marsh had been the cornerstone of Australia's first-innings total, continued his assault by pulling Woakes for six.
Head also hooked two successive Wood deliveries for six before he was last man out, when he holed out off Broad (3-45), the senior England bowler in an attack without the rested James Anderson.
England are bidding to become just the second team in history to win a Test series from 2-0 down after an Australia side inspired by batting great Don Bradman overturned that deficit to take the 1936/37 Ashes 3-2.
