The first concrete chance of the game fell to Colchester's Freddie Sears, whose low strike in the 11th minute smashed the base of the post and rebounded away from goal.
Sears was United's standout player and tested the Walsall defence again a few minutes later with a powerful strike that ended up being blocked.
The away side had a penalty appeal turned down early on after the restart when Sanchez Watt went over in the Saddlers box, but the referee was unmoved.
Aside from the odd U's foray, the game went on to be dominated by Walsall, as the home side saw chances from James Baxendale and Reece Flanagan fly just off target.
Bradshaw arguably had the best chance for the home side at the death, but he fired wide following a defence-splitting pass from Baxendale.