France were competing at a major tournament for the first since they endured penalty shootout disappointment in the 2022 World Cup final against Argentina.
Les Bleus took the lead in the 38th minute as Maximilian Wober diverted the ball into his own net from Mbappe's driven cross.
After failing to score at Euro 2020, Mbappe was presented with a glorious chance to break his European Championship duck when he bore down on goal, only to send his right-foot shot just wide of the goalkeeper's left-hand post.
The France captain ultimately experienced a painful end to the evening, suffering a suspected broken nose following a collision with Kevin Danso.
Mbappe was subsequently withdrawn, leaving Didier Deschamps's side to successfully see out the final stages without their skipper as the 2018 World Cup winners stood firm to claim an encouraging clean sheet and maximum points.
Wober left red faced in Dusseldorf
Unsurprisingly, Mbappe was named in France's starting lineup to make his first appearance since being unveiled as a Real Madrid player.
Mbappe required less than 10 minutes to get his first meaningful sight of goal, as he drove into the box before being denied at the near post by Austria goalkeeper Patrick Pentz.
Austria may have experienced an early scare, but they certainly enjoyed moments of promise in the first period, including in the 35th minute when they carved out a golden opportunity to take the lead.
Marcel Sabitzer produced a cushioned pass from Michael Gregoritsch's lofted cross to tee up Christoph Baumgartner, who opted to try and lift the ball over Mike Maignan, only for the AC Milan goalkeeper to knock it around the post.
Not long after, Ralf Rangnick's side were left to rue their luck as the 2022 World Cup finalists hit the front courtesy of Mbappe's intent and a piece of fortune.
The former Paris Saint-Germain man showed great trickery and pace to drive away from Phillipp Mwene and create room for a driven cross which found the head of Wober, who diverted it into his own net in his attempt to deal with the delivery.
After helping his country establish a half-time lead, Mbappe looked set to put his own name on the scoresheet when he raced through on goal in the opening stages of the second period.
However, in his attempt to pick out the bottom corner, the France captain could only steer his ball wide of Pentz's goal to the shock of everyone in attendance at the Dusseldorf Arena.
Austria then survived another scare when Marcus Thuram found himself in a promising position, only to see his near-post shot kept out by the resolute Pentz, keeping Das Team within reach of their heavyweight opponents.
Despite showing great determination, Austria's attempts to carve out a late equaliser ultimately proved fruitless as they fell to their first defeat since October 2023.
Frustration for Mbappe
Although France will be pleased with the narrow win, the evening proved to be a frustrating and painful one for Mbappe.
The 25-year-old may have played a key role in France's winner, but he would have been disappointed to spurn a golden opportunity to end his wait for a European Championship goal.
France will also be concerned about their talisman after he appeared to sustain a broken nose when his face collided with the shoulder of Danso.
To add insult to injury, Mbappe was shown a yellow card when he came back onto the pitch forcing a substitution which saw him eventually replaced by the experienced Olivier Giroud.
The forward will hope to get patched up and make himself available for Friday's mouthwatering duel with the Netherlands in Leipzig.
As for Austria, they may feel they need to beat Poland in their second group game to have a realistic chance of reaching the knockout rounds for the second European Championships in a row.