Shrewsbury, who have faltered at home with four draws in their last five home games, equalised only seven minutes later with the recalled Jake Robinson and Lewis Neal combining to set up Hibbert for a low shot from 10 yards.
The hosts lost Dean Holden and Craig Disley to injuries in the first half hour but grabbed what proved to be the winner in the second minute of first-half stoppage time when sub Steve Leslie beat Tony Roberts with an overhead kick.
Dagenham had Scott Doe sent off for an elbow on Terry Dunfield after 70 minutes and Shrewsbury played some fine-flowing football but failed to add to their lead.
Source: Team Talk
Source: Team Talk