Grabban's strike four minutes before half-time extended Bournemouth's unbeaten run to 13 games since manager Eddie Howe's return to the club in mid-October.
But the victory was soured by a nasty injury to midfielder Shaun MacDonald, who was carried off shortly before half-time with a serious ankle problem.
Bournemouth found it difficult to penetrate as Colchester, beaten six times in their last seven matches, defended in numbers.
Wes Fogden, Steve Cook and Brett Pitman all went close before the in-form Cherries eventually took the lead after 41 minutes.
Colchester failed to clear their lines from a corner and Shaun O'Kane slipped Grabban in for his ninth goal of the season.
Colchester made three substitutions and eventually fashioned a shot but Josh Thompson was off target.
For the Cherries Marc Pugh became increasingly more of a threat on the left flank and twice set up opportunities wasted by Pitman.