The League Two side more than matched their higher division opponents as they picked up a first victory in six and booked a second-round tie with Preston.
Wanderers took the lead when Scottish striker Craig dusted himself down to score from the penalty spot after 48 minutes after being felled by Alex keeper Ben Garratt.
And then the visitors secured the win after 70 minutes when ex-Norwich defender Gary Doherty slid in to net from close range from a well-worked Josh Scowen courner routine.
The game had looked destined to end goalless after a turgid first half produced few shooting opportunities.
The best of them fell to Wycombe's Paris Cowan-Hall, who twice failed to make the most of good openings, whilst Max Clayton came closest for Crewe.
But after the break Gareth Ainsworth's side stepped up a gear and secured an easy win.