After little action at either end, the visitors took the lead in the 27th minute when Joe Anyon was forced to parry a close-range shot from Joyce leaving him unable to save Danny Graham's tap-in.
Jeff Smith was out to embarrass his former employers and almost did just that in the 32nd minute when he struck a curling effort from distance that Anyon had to tip over the crossbar. Vale's best effort of the first half came a minute before the break when Robin Hulbert produced a snap-shot from 20 yards that flew inches clear of Kieren Westwood's right post.
Martin Foyle's men almost grabbed themselves an equaliser only seconds after the restart when Malvin Kamara turned in the box to squeeze a deflected shot towards goal that fell into the side-netting. Westwood was kept busy once more in the 56th minute when Paul Harsley stole in at the front post with a looping header that the goalkeeper managed to turn over his own crossbar.
It looked like it wasn't going to be Port Vale's afternoon when Akpo Sodje was gifted a prime opportunity to slot the ball into the net from six yards but the striker poked it past the post. The introduction of Luke Rodgers and Nathan Lowndes meant the home side now had four strikers on the pitch, but even the sending off of Joyce in the 81st minute didn't seem to help Vale's cause.
Carlisle completed a smash-and-grab away performance when Joe Garner fizzed a 25-yard stunner off the underside of the crossbar and just inside Anyon's top left corner. The win for the Cumbrians leaving them five points away from the play-off places with five games to go.