Goaltenders Mike Brodeur and Henrik Lundqvist were locked in a scoreless duel when Campoli's third goal this season gave Ottawa a 1-0 lead at 18:46 of the third period.
Kelly's ninth (empty net, unassisted) came 63 seconds later, breaking the Senators' five-game losing streak.
Brodeur was credited with 32 saves on 32 shots.
Mike Brodeur, Team Canada's Olympic goalie, had been recalled from the minors on an emergency basis after starter Pascal Leclaire was injured in practice Wednesday.
Lundqvist was beaten once on 33 shots for the Rangers, who were shut out Wednesday by New Jersey's Martin Brodeur in a 1-0 shootout.
They had won four in a row before the back-to-back shutout losses.