The home side started well enough when Luke Rodgers played a clever ball into David McGoldrick, who held off pressure before firing a low shot into the arms of Shane Higgs.
Cheltenham threatened in the 17th minute, but it was more by mistake when Andy Lindegaard's right-sided cross was mishit and fired into the side of Joe Anyon's goal.
The Valiants took the lead in the 20th minute to the horror of the visiting supporters as McGoldrick's half-effort from two yards was adjudged to have gone over the line, despite the striker looking like he had used his arm to bundle the ball into the net.
The opposite official then made a strange decision of his own three minutes later when his information led to referee Neil Swarbrick sending off both Jason Talbot and Lindegaard for a minor altercation.
Loan player McGoldrick should have had a second eight minutes before the break when George Pilkington floated over a delicate cross, but the striker directed his header wide of Higgs' right post.
Higgs was kept busy in the run-up to half-time as Justin Miller tried his luck from 30 yards, narrowly missing the target with the ball flying inches over the crossbar.
Cheltenham's inability to finish was made apparent in the 57th minute when Jerry Gill failed to test Anyon when he was presented with a perfect shooting opportunity from close range.
Vale continued to push for another and after going close through Rodgers and Richards, Paul Edwards opened his account for the season with a brilliant run and cool sidefoot finish from 12 yards.
A professional home performance was completed three minutes before the final whistle when Shane Tudor's through ball played Craig Rocastle clean through on goal to finish underneath the helpless Higgs.