Vale made a strong start and the visitors' survived some shaky moments as Ben Williamson fired wide and striker Tom Pope gave visiting defender Scott Cuthbert plenty to think about.
Orient remained on the back-foot against their fired-up hosts before taking the lead against the run of play four minutes before the interval.
Moses Odubajo slipped the ball into the path of Cox 20 yards out and the midfielder struck a perfectly-placed right-footed shot which clipped the underside of the crossbar on its way in.
It was tough on Vale, who continued to ask questions at the start of the second-half when Doug Loft's header was cleared off the goalline by Lloyd James.
But Orient doubled their advantage in the 58th minute following a rare slip by Vale goalkeeper Chris Neal.
James's right-wing corner was headed goalwards from 12 yards by Clarke and Neal failed to keep the ball out of the left-hand corner of his net.
Vale continued to show terrific spirit and had the ball in the net twice only to be denied by infringements, while substitute Billy Knott rattled the crossbar.
Orient goalkeeper Eldin Jakupovic was also outstanding throughout the second period as Vale somehow drew a blank.