The hosts had an early penalty appeal waved away when striker Rory Fallon went down following a challenge by John Guthrie before Crewe went ahead after 18 minutes.
Goalkeeper Paul Jones could only parry a low shot from Mathias Pogba and Uche Ikpeazu, on loan from Watford, had the easiest of tasks to steer the ball home from close range.
At the other end Ben Garratt denied Crawley an immediate equaliser by expertly tipping a goal-bound shot from Billy Clarke over the crossbar.
Garratt later showed alertness by coming out to thwart the on-rushing Billy Clarke on the edge of the area.
Crewe's leading scorer Chuks Aneke had a goal-bound shot saved by Jones shortly after the break when a Byron Moore cross was headed back by Pogba.
Garratt cleared confidently at the feet of Clarke before midfielder Mike Jones was narrowly off target for Crawley with a low shot which flashed wide.
The visitors lived dangerously as a deflected shot from Mark Connolly came back off a post and substitute Jamie Proctor then forced Garratt into another smart save.
The Red Devils pulled level after 69 minutes when Proctor let fly with an angled drive which flew into the top corner of the net.
But Inman struck 10 minutes from time to give Crewe only their second win in the last nine games by steering in a Moore pass.