Shrewsbury grabbed their first league win against a 10-man Crewe side still seeking their first point.
Sean McAllister's glancing header from Lionel Ainsworth's cross put the home side ahead.
Crewe substitute Nick Powell was sent off for a studs-up challenge but Alex could have equalised when Max Clayton headed against the post.
Nicky Wroe wrapped up Town's victory with a brilliant solo run in stoppage time.
Crewe remain bottom of the Football League after their fourth straight defeat.
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Shrewsbury Town manager Graham Turner told BBC Radio Shropshire:
"That's a big relief. The confidence of getting our first win will make a difference to the players.
"We played some nice stuff going forward and we got ourselves into some good positions, but we've got to start getting on the end of some of these crosses.
"It's great to get our first win. We've got a week now where we can look forward to a good Carling Cup tie against Swansea."
Crewe manager Dario Gradi told BBC Radio Stoke:
"They got off to a good start but, by half-time, I thought we were back in the game.
"Nick Powell getting sent off obviously didn't help but we hit the post and took the game to them.
"The referee couldn't get the card out quickly enough. He didn't stop to think about it."