The striker tucked home with 11 minutes to go and the Robins sit just outside the play-of places on goal difference.
David Flitcroft’s side had a great chance to open the scoring when Kaiyne Woolery cut inside from the left flank and laid the ball back to Keshi Anderson, but his effort was palmed away by Brad Collins.
Forest Green lost Farrend Rawson to a serious injury midway through the first half, with the defender stretchered off after clashing with Ellis Iandolo.
Anderson was again denied by Collins just before half time, with the goalkeeper diving to his right to turn behind the midfielder’s curling shot.
The visitors started the second half the brighter of the two teams, with Christian Doidge’s low strike forcing Reice Charles-Cook into a save.
The home side then went ahead after Anderson’s shot was parried by Collins and after a goalmouth scramble Norris was on hand to tap the ball into the net.
Colchester will hand late fitness tests to Ryan Jackson and Brandon Comley prior to their clash with Yeovil.
Arsenal secured their place in the quarter-finals of the Europa League following their 3-1 defeat of AC Milan at the Emirates Stadium on Thursday.
Danny Welbeck scored twice against AC Milan to toast his England recall by sending Arsenal into the last eight of the Europa League.
Ipswich striker Joe Garner will not play again this season after it emerged he suffered a fractured skull against Sheffield United last week.