An injury-time header from defender Richard Rose ended Dagenham's run of five straight league defeats.
Gavin Tomlin forced two fine saves from Crewe keeper Steve Phillips while Nick Powell's half-volley was brilliantly saved by Daggers' Chris Lewington.
Crewe went ahead when Shaun Miller's shot was parried by Lewington and Byron Moore pounced at the far post.
Sam Williams turned in Tomlin's cross for the equaliser before Rose headed in his first goal for the club.
VIEWS FROM THE DRESSING ROOM
Dagenham and Redbridge boss John Still told BBC London 94.9:
"The key is to be solid defensively, and that was a platform for us to win the game.
"I thought we played OK, but if fortune favoured us we could have had a couple more goals.
"Their front two is the cleverest pair we have come across, but we were more like Dagenham in the last two weeks."
Crewe Alexandra manager Dario Gradi told BBC Radio Stoke:
"It was very cruel for the players and also for us and the supporters."
"Overall I thought the boys played some decent football, but we've got it in us to concede and make mistakes.
"Dagenham were direct and strong and were decent too - we have to stand up to that."