The former Liverpool and Newcastle striker made a surprising move by dropping two divisions from Ligue 2 side Amiens to sign for Bordeaux last week.
Babacar Leye had Voltigeurs 2-0 ahead at halftime before Carroll's intervention.
The former England international first struck a low volley on 72 minutes, then he scored with a trademark header in the 88th minute to rescue a point for his new club.