The Posh went ahead through a well-drilled set-piece, which saw debutant Shaq Coulthirst set up Erhun Oztumer.
Gillingham levelled when Dominic Samuel broke free and rolled the ball to Norris, who slotted past Ben Alnwick.
Dack completed Gillingham's comeback when his deflected shot beat Alnwick, before Peterborough's Chris Forrester's 35-yard effort struck the post.
Peterborough boss Graham Westley told BBC Radio Cambridgeshire:
"Sixty per cent possession us, 40 per cent them, 20 goal attempts us, 15 attempts, them; I think we played more than enough good football to win the game.
"In the end, it probably was that first-half period when we were right on top and didn't take our chances that has come back to haunt us.
"With a young team, sometime they don't smell the blood and understand how important it is to convert domination into goals.
"We had them chasing shadows for a long period in that first half, but if you don't convert it into goals, a home side - especially a home side that's been going well - is always likely to come back and produce the goods."
Goals from Bradley Dack and Luke Norris saw Gillingham come from behind to beat Peterborough and go top of League One.