He had earlier failed to convert a glorious chance that would have allowed France to win by a wider margin, and sources close to the player confirmed he had broken his nose in the accidental clash with Kevin Danso.
“He is in a bad way. He is not well. His nose is a mess, that’s for sure. That is the black mark of the evening for us,” France coach Didier Deschamps said, without confirming the extent of the injury.
“We had the chance to put the game to bed and it is true that we could have been more precise in attack, but it’s very good to start with a victory,” Deschamps said of the result.
Despite Mbappe’s injury, France will be relieved to have emerged victorious after a build-up overshadowed by talk of politics rather than football.
Mbappe and other French players had spent much of their time before the media in recent days answering questions about upcoming elections in the country which could see the far-right National Rally become the biggest party.
Any slip-up here would have led to suggestions their minds had not been fully focused on the competition.
Instead, the result leaves France level on three points in Group D with the Netherlands, who defeated Poland 2-1 on Sunday.
France and the Netherlands meet next in Leipzig on Friday, while Austria face Poland the same day in Berlin.
“We are obviously disappointed with the result. No France player scored against us in 100 minutes, just a very unfortunate own goal,” said Rangnick.
“I think Didier Deschamps was really happy when the final whistle came which shows how well my team played.”
Austria had enjoyed some impressive results coming into the tournament and their team is on familiar ground during these Euros — Rangnick is from Germany and eight of their starting line-up played in the German Bundesliga last season.
Woeber own goal
The French, however, are arguably the most talented team at the tournament, led by their captain Mbappe.
He endured a miserable time at his only previous Euros in 2021, failing to score and missing the decisive penalty in a shoot-out defeat by Switzerland in the last 16.
The new Real Madrid signing will wonder how he did not score in this game, with his first big opportunity arriving inside eight minutes.
Antoine Griezmann and Theo Hernandez combined to release Mbappe, who cut in from the left and shaped to curl a shot into the far corner before instead aiming for the near post.
However, Austria goalkeeper Patrick Pentz managed make the save.
The Austrians settled into the contest and passed up a wonderful chance of their own to go ahead on 36 minutes.
Skipper Marcel Sabitzer touched down a Michael Gregoritsch cross from the left for Christoph Baumgartner, but he was foiled by goalkeeper Mike Maignan.
That proved crucial as France took the lead two minutes later.
Ousmane Dembele gave the ball to Mbappe on the right, and he produced a stepover to get to the byline before his cutback was nodded into the far corner of his own net by the unfortunate Woeber.
Mbappe was then thwarted by Pentz as he tried to go around the goalkeeper in first-half stoppage time, but that was nothing compared to the chance he squandered 10 minutes after the restart.
Adrien Rabiot sent Mbappe away, bounding in behind the Austrian defence, but he somehow put his shot wide with just the goalkeeper to beat.
That left the game in the balance, and Austria continued to push for a leveller, without success.
Their fans were less than impressed by what they perceived as timewasting late on from Mbappe, who needed treatment after colliding with Danso in the box.
He left the pitch with his shirt stained by blood, only to then come back on and sit down, earning himself a yellow card before being replaced by Olivier Giroud.
© Agence France-Presse