Vale made a bright start and almost took a ninth-minute lead when Pope's header thudded against the crossbar from a Chris Lines corner.
And Vale's dominance was rewarded in the 32nd minute when Jennison Myrie-Williams fired a penalty into the top corner of Chris Day's net after Ben Chorley had fouled Gavin Tomlin.
But Stevenage woke from their slumber to score in both of their first two serious attacks shortly before half-time.
Jordan Burrow guided a right-footed shot low inside Sam Johnson's post after Francois Zoko skinned Chris Robertson to cross from the left.
And Burrow turned to smash his second into the roof of the net with a left-footed effort from six yards after Zoko again caused havoc in the Vale defence.
Stevenage were reduced to 10 men in the 71st minute when Michael Doughty, who had earlier been booked for time wasting, was shown a second yellow card for a foul on Carl Dickinson.
And Pope finally rewarded increasing Vale pressure when he met Rob Taylor's left-wing cross with an unstoppable eight-yard header just inside Day's left-hand post.