Danny Philliskirk had a chance to get Oldham back on level terms later in the first half, but his strike was deflected behind for a corner, which was swiftly cleared.
The home side continued to apply the pressure late in the first half, but Sam Johnson made a string of saves to keep Port Vale ahead.
Chris Lines came close to restoring Port Vale's advantage when he headed the ball onto the cross bar following a Pope flick forward.
Johnson had to be on hand late on as Oldham won a corner in the final five minutes, but the keeper claimed a headed effort to keep the scores level and secure the draw.
The draw gives Oldham their first point of the season, moving them up to 21st in the table, while Port Vale sit ninth.