Neumann's 13th-minute header put Birmingham in front at the Racecourse Ground before Callum Doyle levelled early in the second half.
Birmingham finished stronger and Neumann nodded in the winner just after the hour to leave Wrexham on two points from three league games.
Birmingham urgently needed a response from Tuesday's 6-1 League Cup thrashing at home to Brentford and delivered to give Davies some breathing space.
It was Birmingham's first Championship victory this season after draws with Sheffield United and Bristol City in their opening two games.
Birmingham are unbeaten in their five meetings with Wrexham since the fortunes of the two clubs were transformed by their celebrity owners.
Wrexham have climbed from non-league obscurity to the second tier thanks to the investment of actors Ryan Reynolds and Rob Mac, while Birmingham are backed by a consortium including NFL legend Tom Brady.
