The 24-year-old took advantage of a tepid fourth-day pitch in Antigua as he reached an unbeaten 117.
Crawley shared a 193-run partnership with captain Joe Root as England ended day four on 217/1. This leaves the tourists 153 runs ahead with three sessions of the match remaining.
This felt like an important personal triumph for Crawley who has been on a rollercoaster journey since breaking into the national side 19 months ago.
The youngster kicked off his international career by blazing 267 against Pakistan, but only averaged a mere 10.81 throughout a disappointing 2021.
He was inevitably dropped late last summer, but has since returned and is beginning to make serious in-roads.
Speaking after his century, Crawley said: "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.
"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.
"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."
It's been 21 innings since Crawley last celebrated a hundred, with 12 of those seeing him score less than 10.
During that time, he was too often guilty of flashing outside off stump, placing too much confidence in flowing cover drives that look glorious when they come off and calamitous when they take the edge.
On this occasion though, the 24-year-old was calculated and chose his moments carefully. What's more, in Root he found the perfect partner.
He added: "I like to put some pressure on the bowlers but I've certainly tried to rein my game in a bit more since coming back into the team.
"I've tried to put away a few more shots that I was playing last year and getting me into trouble. That's what I've been trying to do: just make the game a bit more simple.
"Whenever I bat with Joe he talks about making good decisions for long periods of time. He does that extremely well and I tried to emulate him. He was very good to bat with, he played brilliantly and took the pressure off me.
"He's always a calm head telling me to take it one ball at a time, make good decisions and thankfully I made a few more than I usually do."
Having reduced their prospects of defeat to almost nil, forcing victory remains a tough ask. Nevertheless, England and West Indies will resume the opening Test at 14:00 GMT.