The sides were promoted together from League Two last season but it is Posh who have adapted better to the higher grade and they took the lead with just three minutes played.
George Boyd played the ball over the top of the defence and Aaron McLean seized on the pass, rounded goalkeeper Peter Gulacsi and slipped home his 16th goal of the season.
Hereford fired in a couple of long-range efforts through Steve Guinan and Sam Gwynne, but it was the visitors who looked more likely to find the net again.
On nine minutes, McLean drove a shot across goal and wide.
The powerful striker was then denied by an excellent Kris Taylor block when he sought to pick up the pieces after Gulacsi had parried a fierce drive from skipper Russell Martin.
McLean was the threat again on 27 minutes firing wide from a great position.
The visitors squandered an excellent opportunity to increase their advantage just two minutes into the second period.
Boyd drove in a cross from the left to pick out Craig Mackail-Smith, but the Posh top scorer squirted the ball wide from just four yards out when it looked easier to score.
Guinan flashed a volley inches wide of the target, but Hereford found it hard to find a way in to test the commanding Joe Lewis in the visitors' goal.
In contrast Gulasci had a number of shots to deal with, but he was fortunate with just six minutes to play when substitute Chris Whelpdale hit a shot straight at him with the whole of the goal to aim at.