Crewe moved above Plymouth and off the bottom of League Two after claiming their first victory of the season.
Shaun Miller converted a cross from AJ Leitch-Smith to put the Alex in front just before half-time.
Defender Harry Davis almost doubled the advantage, but his goal-bound header was kept out by team-mate Miller.
Visiting goalkeeper Steve Phillips was rarely tested, with Carl Fletcher and Simon Walton coming closest to an equaliser with long-range efforts.
VIEWS FROM THE DRESSING ROOM
Plymouth manager Peter Reid told BBC Radio Devon:
"It's tough times, I thought there was a lot of anxiety among the players. Today I thought we were nervy, edgy and it certainly showed in our play.
"It was a really disappointing result. We didn't pass it as well as I would have liked.
"Too many times our players got isolated and that's a desire to work and get angles for your mates and we didn't do it well enough."
Crewe manager Dario Gradi told BBC Radio Stoke:
"It's a great relief, for the players and supporters as well, not just for me. We'll travel home happy but there's still a lot of work to do.
"The goal was an outstanding bit of play by the front two, who worked very hard without much success.
"I thought we defended well from the front backwards, which we've worked on. They hit a lot of long balls which is what we wanted them to do."