The striker swept the ball home from inside the area after the hosts had failed to cut out a ball from the right to leave them still without a win this term.
Lee Miller came within inches of giving Carlisle a third-minute lead when his header from James Berrett's cross came back off the bar with goalkeeper Chris Neal beaten.
However, they were very nearly exposed by Richard Duffy's diagonal ball which found Pope in acres of space just outside the penalty area. He drilled a shot low but Mark Gillespie was quick to get down and make the save.
One of Carlisle's three debutants, on-loan Swindon defender Troy Archibald-Henville, missed a great chance three minutes before half-time, heading Berrett's corner over the top from inside the six-yard box.
Danger signals flashed for the hosts straight after the break when Louis Dodds' shot on the turn looked to be going in until Danny Livesey cleared off the line.
The Cumbrians should have grabbed the lead inside the last minute of normal time when Brad Potts snatched at a chance five yards out and poked his shot over the bar.
It proved costly too as Pope stepped up to give Vale the win.