Vale took the lead on 50 minutes when they broke straight from a Town free-kick before Turner chipped the ball over the onrushing Russell and into the back of the net.
But the Mariners were soon level after Rose was tripped in the box and Cook stepped up to slam home.
Vale were to restore their lead when Miller’s shot from distance took a huge deflection to send it looping over a helpless Sam Russell.
Miller added the visitors third from the penalty spot late on.
Town: Russell, Davis, Trialist, Öhman, Hall-Johnson, Woolford, Welsh (c), Cook, Ring, Burrell, A. Rose.