Former Ipswich midfielder Drury struck in the 64th minute when he headed in from five yards after being set up by Jamie Proctor.
Aaron Wildig had put Shrewsbury ahead a minute after the break by volleying in from 15 yards following a cross from Ryan Woods.
Shrewsbury striker Curtis Main later headed wastefully over from close in and also blazed narrowly wide as honours finished even.
Crawley's top scorer Billy Clarke wasted a golden chance to put his side ahead from the penalty spot two minutes before the break after Darren Jones had brought down Nicky Adams.
Shrewsbury goalkeeper Chris Weale dived to his right to push the Irishman's spot kick on to the post.
Clarke had earlier had a goal-bound shot blocked by Jones following good work by Adams.
Shrewsbury threatened in the sixth minute when Jon Taylor's firm shot was parried by home goalkeeper Paul Jones and Tom Bradshaw skewed the rebound wide.
Adams wastefully fired an angled shot off target before Shrewsbury midfielder Paul Parry forced Jones into an important block from his low goal-bound shot.