Callum Doyle and Armstrong both had efforts on target early on in what was a fairly low-key opening to the contest in north Wales.
Rob Mac was delayed in linking up with fellow Wrexham co-owner Ryan Reynolds at the Racecourse Ground, although, upon his arrival, the game exploded into life after Mateus Mane was well denied by Anthony Patterson.
Moments after that chance, Nasser Djiga’s long ball upfield caught out the Wrexham defence, and Armstrong smartly volleyed beyond Patterson to open the scoring with his fifth goal of the campaign.
That briefly took Wolves to the top of the table, although the joy of the away supporters was short-lived.
Issa Kabore’s lofted cross from the right found O’Brien who nodded beyond Jose Sa from six yards out – the midfielder’s first goal since January.
O’Brien and Ollie Rathbone both had efforts either side of Marshall Munetsi’s wayward shot as the contest remained finely poised.
Peixoto’s side lifted the tempo after the restart as Mane drilled into the side-netting before Fer Lopez blazed over the crossbar moments later.
The hosts spurned a glorious chance to go ahead after the hour mark as Jacques Ekomie’s free-kick fell to Kieffer Moore, who failed to guide his effort on target.
Patterson made further saves to deny Lopez and Tommy Doyle as the visitors looked the likelier to net a late winner.
But Wrexham proved a tough nut to crack as the Welsh side earned a point to ensure they go into the international break in positive spirits – with Wolves still firmly in the mix for a top two spot.