The visitors were never in the game and Derby could have scored more goals but for Ross Turnbull, who made several excellent saves.
Crewe came into the match after the club's training ground was shut down when goalkeeper Ben Williams was taken to hospital with suspected viral meningitis.
Manager Dario Gradi refused to put that forward as an excuse because it was not the ideal preparation for a team who had not won since early November.
They never looked like breaking that depressing run as Derby controlled the game from start to finish although the score line failed to silence fans who protested against the board of directors.
At least the players got it right by taking full advantage of a flimsy Crewe side to ease their own relegation worries.
Paul Peschisolido scored his thirrd goal in eight days after Turnbull had blocked a Danny Graham header and defender Michael Johnson headed a second in the 21st minute from another Inigo Idiakez corner.
Crewe avoided further damage before half-time but they were swept aside after the interval despite the agility and bravery of Turnbull.
Tommy Smith re-discovered his scoring touch by sliding a low precise shot to Turnbull's right after Graham had sent him clear in the 53rd minute.
Smith struck again ten minutes later when Graham was denied twice by Turnbull, who could do nothing as the loose ball was smashed into the unguarded net.
Smith was denied a hat-trick by a spectacular save, but the keeper was shown a yellow card when he kicked the ball away in frustration after the referee decided a defensive interception was a backpass.
The free-kick from eight yards was lifted into the roof of the net by Idiakez and although David Vaughan replied with a 25-yard dipping shot 11 minutes from time it did not alter a miserable day for the visitors.