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Peterborough gave the Valiants a scare in the 19th minute when Conor Washington fired a dangerous shot straight at Chris Neal, with the rebound falling to Luke James who went on to blast over the bar.
The home side did break the deadlock on the half-hour mark through Marcus Maddison. The signing from Gateshead collected the ball on the right wing following in a throw-in and turned to lash the ball into the top corner of the Vale net.
Peterborough doubled their lead midway through the second half when a Jon Taylor free kick in the 71st minute fell to Kyle Vassell, and the young striker duly headed the ball into the back of the net.
Posh were not finished there and got on the scoresheet once again just seven minutes later courtesy of Jack Payne's ambitious 30-yard strike, which flew into the Valiants net and sent the home fans into raptures.