Leicester loanee Gallagher's unstoppable 25-yard effort into the top corner of the net was sandwiched by Joe Garner's double as North End secured their third consecutive win despite Gavin Tomlin and Jordan Hugill pulling goals back for the visitors.
Garner headed Preston in front after 43 minutes, nodding Lee Holmes' cross beyond Vale goalkeeper Sam Johnson from close range.
Gallagher doubled the lead five minutes into the second half, receiving the ball in space before unleashing a stunning shot to give Johnson no chance.
Garner added a third goal after 62 minutes when he shot high into the net from Gallagher's unselfish pass after Iain Hume's close-range header was parried away.
Tomlin's diving header seven minutes later from Jennison Myrie-Williams' cross gave the travelling supporters some hope as Vale piled the pressure on.
And with six minutes remaining substitute Hugill headed a second at the near-post, but Simon Grayson's side clung on to seal all three points.