The Seasiders went into the game two points adrift of sixth-placed Swansea but they leapfrogged them due to the Swans' 2-0 defeat at Sheffield United.
The Tangerines piled forward from the start and gave Peterborough an early warning when in-form striker Campbell beat debutant Danny Andrew before firing just wide.
The visitors continued to exert pressure and Brett Ormerod could have given them the lead but the forward shot wide.
Ian Holloway's men took a deserved lead in the 12th minute through Campbell although there were suspicions of offside.
Charlie Adam drove forward and found Stephen Crainey who whipped an excellent cross on to the head of Gary Taylor-Fletcher. The striker then headed into the path of Campbell who forced a save from James McKeown with an acrobatic overhead kick before finishing the rebound.
Spurred on by their opener, the Seasiders pushed for a second and nearly grabbed it but McKeown produced an excellent save to deny Ian Evatt from David Vaughan's corner.
Peterborough started to work their way back into the game as Liam Dickinson struck wide from 18 yards.
It was Blackpool who created the game's next real chance, though, when Taylor-Fletcher found himself clean through before squaring to Campbell who forced another good save from McKeown.
Whatever Gary Johnson said to his players during the interval seemed to work as Posh started to threaten their play-off chasing opponents.
And they should have restored parity but Craig Mackail-Smith saw his effort cleared off the line before Nathan Koranteng hit a great chance wide.
The visitors' struggles early in the second half were summed up by the form of star-man Adam who was clearly rattled by being a yellow card away from a three-match ban.
Blackpool still looked dangerous when attacking the hosts' fragile defence, though, as Ormerod forced a save from McKeown.
The noise created by the travelling faithful went up a few decibels when news of Richard Cresswell's opener at Bramall Lane filtered through and the Tangerines responded by piling forward in search for a second.
And they should have got it in the 75th minute but Campbell spurned a fantastic opportunity to grab his eighth for the club.
The hosts might already be relegated but they were not going down without a fight as they desperately pushed for a leveller.
And they had a late shout for a penalty when Dickinson was brought down by Adam but referee Colin Webster waved play on as Blackpool saw the game out.