Moore set Crewe on their way in the 27th minute with a cool finish against the run of the play before Inman doubled their lead just after the hour as Preston crashed to a third successive defeat.
Inman curled home from outside the penalty area as Crewe got back to winning ways having lost their last two on the road 4-0 and ensured Preston's poor form after five straight wins continues.
Preston could have gone ahead after two minutes, but Scott Laird's own curling effort was well saved by Crewe goalkeeper Steve Phillips with Iain Hume firing an effort on to the crossbar from the resulting corner.
Stuart Beavon drew another save from Phillips after 20 minutes before Moore tucked home Chuks Aneke's clever chipped through-ball moments later to give Crewe the lead.
Kevin Davies headed Hume's 59th-minute cross agonisingly wide before Crewe broke to score a decisive second through Inman, while the visitors almost grabbed a third when Adam Dugdale hit the crossbar soon after.