Dario Gradi has been installed as caretaker manager at Gresty Road after the sacking of Steve Holland during the week but the 67-year-old was unable to improve his club's fortunes, despite a lively first-half showing.
Former Crewe keeper Owain Fon Williams denied on-loan midfielder Stuart Green, who had cracked an early volley goalwards.
The Wales Under-21 international got in the way of a John Brayford effort and Joel Grant clattered the woodwork with a 20-yard shot as Crewe offered a genuine threat.
But Jim Gannon's Stockport side controlled much of the game and former Wales international striker Craig Davies was an invaluable target man.
Davies had the ball in the net just before the half-hour mark when he netted after Steve Collis had parried Tommy Rowe's shot.
But the County striker wasn't to be denied three minutes into the second half when he latched on to a ball on the right flank, turned his marker Danny O'Donnell and then forced the shot under Collis at the near post.
Pilkington edged Stockport further ahead in the 66th minute, when he twisted and turned before lashing a low finish into the bottom corner.
Crewe struggled to respond and Gannon's side finished the game passing the ball around their opponents.
Both Pilkington and Davies came close to grabbing their second goals of the game but it was substitute Thompson who rubbed in the misery for Crewe, in stoppage time.
When Dominic Blizzard hit Leon McSweeney's cross back across the face of goal, the Northern Ireland striker stooped to head home a close-range finish.
Gradi said after the defeat: "We were full of life in the first half, but in the end they outplayed us. I thought we crumbled a bit mentally and we had tired legs out there.''
County boss Gannon enthused: "We played some great stuff today and Craig's goal was the turning point. I'm delighted to have him here on loan, he was a real threat.''