The Shrews should have been ahead early on but Jake Robinson missed a sitter when put through by Dave Hibbert, side-footing wide with the goal at his mercy after rounding goalkeeper Dean Bouzanis.
Accrington had three penalty appeals turned down in the first-half and goalkeeper David Button saved Shrewsbury from further damage.
The former Spurs stopper thwarted Andrew Proctor and Luke Joyce while Grant and Michael Symes missed good chances from inside the area.
Proctor was sent off seven minutes from the end for a second bookable offence.
Source: Team Talk