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.