Nelson scored the winner at the ground where the Canaries were relegated last year, but the visiting side had goalkeeper Fraser Forster to thank for a string of superb second-half saves.
Charlton midfielder Nicky Bailey had the best chances for the home side, who are at least guaranteed a place in the play-offs after Colchester's loss at Hartlepool.
And Addicks striker Nicky Forster was twice kept out by his namesake between the posts.
Both sides got off to a nervy start - indicative of what was at stake for both clubs - but Simon Lappin's inch-perfect corner was met by Nelson's head to cue wild celebrations from the visiting fans.
Deon Burton could have equalised on the hour, but he was also denied by a wonderful Forster save. The visitors packed men behind the ball for the final 20 minutes and held on enough before venturing onto the pitch at full-time to celebrate with the packed away end.