The Rams were reeling when centre-half Andrew Davies was red carded for his second bookable offence when he mistimed a tackle on home striker Eddie Johnson on the hour mark.
But Crewe missed several good chances to seal the three points in the final third of the game. Luke Carney was guilty of the worst miss when he shot straight at Lee Camp when clean through.
Crewe had taken a first-half lead somewhat against the run of the play but Phil Brown's side stroked the ball around pleasingly on a good surface at Crewe.
The Rams had nothing to show for their early dominance bar a Marcus Tudgay header which flew just over the bar. Home keeper Ross Turnbull did fumble a Tommy Smith cross-cum-shot and was lucky to escape when the rebound was deflected wide for a corner.
Derby were even more lax with their defending in the 29th minute when Kenny Lunt whipped the ball across the box leaving Mark Rivers with a simple close-range glancing header to give Crewe the lead.
The Cheshire side were close to a second four minutes later when Lunt curled a delicious free-kick into the box and Steve Foster's powerful header was acrobatically palmed away by Camp from his far corner.
Derby hit back within three minutes of the restart when Danish winger Morten Bisgaard picked up a loose ball just outside the penalty area and produced a stunning low finish just inside the post.
Smith, who was a constant thorn in Crewe's flesh, nearly edged the visitors in front when he got through on goal via Tudgay's pass only to shoot straight into Turnbull's hands.
Inigo Idiakez just missed out on emulating Bisgaard's earlier strike with a similar shot from outside the box that flew inches wide.
But then the game turned when referee Ian Williamson appeared to have no choice but to dismiss Davies who had been booked earlier for deliberate handball.
The decision took the wind out of Derby's sails and Crewe pressed hard for maximum points. After Varney's miss Johnson also came close but Camp again showed how good a goalkeeper he is keeping out a low shot at the near post.
Substitute Steve Jones could have done better it seemed with four minutes left but his shot on the turn was a couple of yards wide.