Crewe have now gone six games without victory, but never looked like ending that sequence after being totally dominated by the Latics at Gresty Road on Tuesday night.
Harry Davis turned Gregor Robertson's corner onto the post with six minutes gone but after that Crewe spent the first half on the backfoot.
Latics striker Derbyshire was denied on the goalline by Kelvin Mellor before Alex goalkeeper Alan Martin pulled off a stunning save to prevent Lee Croft dispatching Christian Montano's cutback.
The visitors finally found the goal they deserved just after the hour-mark when Tarkowski headed home Croft's corner at the far post.
Derbyshire put the result beyond doubt with 11 minutes to go, turning in Montano's superb low cross from close range before Crewe midfielder Luke Murphy hit the crossbar with a free-kick in injury time.