Posh started the brighter of the two teams and spurned two opportunities to take the lead early on, first with Britt Assombalonga narrowly missing the right-hand corner and then his strike partner Barnett thrashed wide from a good angle minutes later.
Darren Ferguson's men continued to threaten, as Assombalonga once again came close to opening the scoring, firing over the bar from close range.
However, the Valiants came close to breaking the deadlock against the run of play moments before the interval, as Lee Hughes found himself with a glorious opportunity before blazing over the bar from inside the area.
The visitors forced the issue after the interval, only to run into the same problems in front of goal that plagued their opening 45 minutes, as Barnett's effort from inside the area lacked the direction to trouble goalkeeper Chris Neal.
Micky Adams's men struggled to create clear-cut opportunities, as Posh enjoyed lengthy spells of possession, which would eventually pay dividends.
Barnett received possession deep in Vale's half and dispatched a right-footed effort past Neal to seal the points for Ferguson's men and keep the pressure on the O's at the summit of England's third tier.