Crewe missed the opportunity to go second in the Second Division when Plymouth's debutant David Norris struck the only goal of the game in first-half injury time.
With Cardiff and Oldham losing, Crewe had the ideal opportunity to make headway but were poor in the first half and unable to break down Argyle's defence in the second.
Norris was signed by Plymouth boss Paul Sturrock from Bolton on loan for a month and he came up with a darting run past Steve Foster to fire in from 20 yards off the post.
It was no more than the visitors deserved for they took an early grip of midfield with Frenchman David Friio outstanding.
Ian Stonebridge and Blair Sturrock tested Clayton Ince early in the game, but Rodney Jack wasted a gilt-edged chance for Crewe when he blasted over from close range after David Wright's long flow had fallen into his path.
The West Indian also tried an acrobatic overhead kick which just dipped over the crossbar.
It was one-way traffic after the break as Crewe swarmed into the Plymouth box time and time again only to be thwarted by poor finishing and some unlucky breaks.
Midfield Dave Brammer stroked a shot against the bottom of the post and centre forward Rob Hulse squandered three good openings. One was a shot on the turn after Jack's cross from the right, which was blocked by a last-ditch lunge from David Worrell.
Hulse also blazed wide in the closing minutes and Steve Jones had a diving header saved by Romain Larrieu.
Crewe manager Dario Gradi admitted: "The players didn't lose their discipline and kept trying right until the end, but perhaps this result shows that we are not quite as good as we thought we were."