The visitors were unable to hold on to the half-time advantage given to them by Courtney Meppen-Walter's goal six minutes before the break.
Vale made the early running and should have taken a 16th-minute lead when Williamson headed over from Jack Grimmer's cross.
But Carlisle gradually got into the contest and it took a decent save by home goalkeeper Chris Neal to keep out a 37th-minute shot from Lee Miller.
Neal was left helpless two minutes later, however, when Courtney Meppen-Walter buried a 10-yard header inside his near post after James Berrett floated over a free-kick from the right flank.
Carlisle looked likely to extend their lead in the early stages of the second period as Neal made fine saves to deny Danny Redmond and Liam Noble.
But the game turned in the 56th minute when Williamson drove a low, right-foot shot inside Jordan Pickford's right-hand post from 12 yards out after cutting in from the left side of the penalty area.
Jennison Myrie-Williams then struck a post before crossing from the left for Pope to head high into the roof of the net after 61 minutes.
Pope had a 77th-minute header tipped against the woodwork by by Pickford and Miller squandered a clear chance to earn Carlisle a point late on.