The 19-year-old attacker scored his side's first goal in the Group B clash at the Veltins-Arena midway through the first half, and soon after the break he won his side a penalty.
However, between being fouled by Garry Bocaly – who received a red card for his clumsy challenge – and colliding with goalkeeper Geoffrey Jourdren, Draxler's left wrist was twisted badly and perhaps even dislocated.
Draxler had the wrist operated on after the match, and is likely to be out of action for the rest of the month.
"If all goes well, it will take a break for three to four weeks (to heal)," said team doctor Thorsten Rarreck, adding that the player could then resume training with a splint on the wrist.
Draxler is set to miss the trip to Bundesliga champions Borussia Dortmund and the Champions League away game at Arsenal.
Klaas-Jan Huntelaar converted the penalty as Draxler was being taken away on a stretcher, but Souleymane Camara's 90th-minute leveller ensured a share of the points for French champions Montpellier.