Home substitute Mark Beck spurned a great chance to grab a late winner, and just as was the case when the sides met at the CNG Stadium at the end of September, neither team could find a breakthrough in a closely-contested affair.
After a slow start, the contest exploded into life in the eighth minute.
Joe Leesley whipped in a cross from the left that eventually fell for an unmarked George Thomson at the far post.
His powerful effort was well-saved by visiting keeper Rob Lainton, before a last-ditch block thwarted Aaron Williams as he followed-up six yards out.
In a role reversal, the resulting corner was curled towards goal by Thomson where Leesley met it at the near post, glancing a header against the cross-bar.
Town continued to threaten and, with 17 minutes on the clock, Leesley cut in from the left and hung up a cross that Williams beat Lainton to, only to nod over the bar.
Moments later, Liam Kitching helped Leesley's corner into the path of Jordan Thewlis, however Lainton extended a hand to brilliantly push over the Harrogate striker's header at point-blank range.
Wrexham's brightest moment arrived when Luke Young drilled a low strike into the bottom corner of the home goal, but the whistle had already gone following a foul in the build-up.
The Dragons had the ball in the back of the net for a second time four minutes after the resumption as Mike Fondop-Talom lifted delicately over James Belshaw when one-on-one, though a late offside flag cut short the visitors' celebrations on this occasion.
The second period lacked the goalmouth action of the first, with both defences generally on top.
Away captain Shaun Pearson did manage to climb and meet a free-kick into the Harrogate area, but directed his headed effort too close to Belshaw.
Probably the clearest chance of the entire game then arrived in the 92nd minute, but home substitute Beck failed to capitalise.
The big frontman somehow nodded a beautiful Leesley cross from the left over the bar just a couple of yards out, meaning the teams must do battle once again in a replay on Tuesday evening.