The visitors almost took an early lead but Graham Cummins' first-time shot was kept out by the feet of goalkeeper Alan Martin.
Inman then audaciously flicked the ball over Paul Connolly's head and unleashed a rasping volley which Declan Rudd did well to save at his near post before the keeper had to palm away Mark Ellis' header.
Preston went close twice but Martin dived low to deny Cummins and then Joe Garner in quick succession.
But Crewe broke the deadlock in the 32nd minute when Kelvin Mellor cutback for Inman to fire home from 12 yards.
Garner swung a shot over the bar after being picked out by Jeffrey Monokana while Billy Wright headed another chance off target and Keith Keane had a long-range effort saved as the visitors looked to level.
Lee Holmes then cracked the post from 25 yards and Jack King's follow-up header was saved by Martin with five minutes to go as Crewe held on to claim all three points.