Crawley (117) and England skipper Joe Root (84) scored freely for an unbeaten 193-run second-wicket partnership, pushing England to 1-217 in North Sound on the island of Antigua.
Stumps were drawn early due to a late afternoon tropical downpour with the visitors 153 runs ahead on Friday.
While a draw is still probably the most likely result, the partnership all but eliminated any chance of a West Indies victory barring an extremely bold declaration by Root early on the final day.
England, however, will be disappointed to have lost more than 23 overs because of the rain.
After West Indies scored at a snail's pace in eking out a 64-run first innings lead, England quickly wiped out the deficit, playing with intent rather than simply trying to occupy the crease.
Opener Alex Lees (6) must be kicking himself after going cheaply for the second time, again trapped lbw by Kemar Roach by a ball that angled back and was destined for leg stump.
The debutant's departure brought Root to the crease and a Test that had been tipping ever so slightly the host's way quickly tipped in England's direction as runs began to flow as freely as the local rum.
Crawley, strong off his legs but not afraid to score on the off side either, completed his century in 181 balls.
His previous three-figure score was a mammoth 267 against Pakistan in Southampton in 2020.
But he had scored only scored 330 runs in 21 subsequent innings before Friday's knock, surely testing the patience of the England selectors.
"It means an awful lot to score a hundred. It's been a while since my last one and there were times last year when I didn't think I'd get another," he said.
"But when I got taken out of the team, they said I had a big future, which I was very thankful for. It gave me a lot of confidence. I was thankful they picked me again.Â
"I always believed in myself that I would come again. Maybe not so soon, but I knew I was young and I had a chance to come again. I feel in good touch now, so hopefully I can make the most of it."
Earlier, West Indies were all out for 375 in their first innings.
The hosts added only two runs to their overnight score before losing their final wicket when Jayden Seales was out lbw for a duck to spinner Jack Leach on the third ball of the day.
Nkrumah Bonner top-scored for West Indies with a marathon nine hour-plus 123, while captain Kraigg Brathwaite (55) was the only other player to reach a half century.
The England bowlers shared the spoils, with Ben Stokes, Craig Overton and Leach each getting two wickets.
The match is the first in a three-Test series in the Caribbean, where England have enjoyed only one series victory in the past half century.
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