Mbappe, who came on as a substitute after the break, found the back of the net with a shot into the top corner seven minutes from time to cancel out Andreas Weimann's first-half opener.
Second-placed Austria and third-placed Croatia have four points each.
Mbappe had a great opportunity to give France all three points with three minutes left, only for his attempt to rattle the crossbar.
"I'm disappointed that we didn't score more with all the chances we had," France coach Didier Deschamps said.
"Sometimes we score more goals with less opportunities but I'm not worried. I've had to do with minor injuries and fatigue."
After an initial 1-0 defeat against Denmark at home and a 1-1 draw in Croatia, Deschamps again reshuffled his team as he looked to rest some key players at the end of a long and demanding season.
In a lively start, France came close in the 18th minute when Griezmann's free kick was deflected in the path of Benzema, whose header was superbly stopped by Patrick Pentz.
But Austria struck first, in the 37th minute, when Weimann tapped in from close range from Konrad Laimer's cross at the end of a sharp counterattack.
France had another good opportunity on the stroke of half-time, only for Benzema's curled shot to be parried away by the impeccable Pentz.
Deschamps replaced Griezmann on 63 minutes with Mbappe, who threatened immediately.
Just as it seemed France had run out of energy and ideas, Mbappe collected a short pass from Christopher Nkunku before beating Pentz from inside the area with a clinical, brutal shot in the 83rd minute.
He came close to doubling the tally four minutes later when Pentz deflected his attempt onto the bar after a one-two with Benzema.
"That's not what we came for but we'll take that point and try to win on Monday," Mbappe said.
"I'm not 100% fit but I'll play if I have to. It's the last match before the holidays so we can push it a little bit."