Vale took an early look at goal when Craig James first went close with a 20-yard free-kick and then later in the ninth minute had a long-range drive saved by the alert Michael Ingham.
Wrexham's only threatening shot of the first half came after 16 minutes when Lee McEvilly turned before diverting Darren Ferguson's cross on to Mark Goodlad's post from eight yards.
The home side got the goal they deserved when, after going close with a header seconds before, Michael Husbands followed up on Nathan Lowndes' parried shot from close range in the 20th minute.
Ingham was called into action again three minutes before the half-hour mark when Lowndes' clever lay off provided Mark Innes with a shooting chance but the goalkeeper got down quickly again to save.
Vale's dominance resumed after the break but poor finishing denied the lead being furthered with both Husbands and Leon Constantine shooting wide of the mark within five minutes of the restart.
Wrexham finally broke the pressure after 63 minutes when Jon Walters' flick on played McEvilly clean through and he made no mistake in slotting the ball underneath Goodlad.
However, their joy was shortlived as after James' right sided corner caused havoc, the ball finally fell for new signing Constantine who squeezed the ball into Ingham's bottom right corner from 12 yards.
The visitors kept pressing right until the end but it was a mistake by George Pilkington that almost levelled the scores when his failed pass resulted in Robbie Foy receiving the ball but Goodlad stood firm to save and send his team through.