Striker Antoine Griezmann lashed home the historic spot kick in the 58th minute but Australia hit back with a penalty of their own when Samuel Umtiti inexplicably handled the ball in the box four minutes later.
Socceroos skipper Mile Jedinak sent his French counterpart Hugo Lloris the wrong way from the spot to give Australia hope of forcing a draw at the Kazan Arena only for Pogba to grab the winner in the 81st minute.
"The first match is always a big one and we're very happy to have won today," said France coach Didier Deschamps.
"It was a difficult match, it was complicated and the Australian team were very good. We were not quick enough in our transition from defence to attack and that favoured Australia."
Even Pogba's goal might not have stood in the days before technology was brought into the game with the midfielder's shot looping off the shin of Aziz Behich, hitting the crossbar and bouncing down inside the goal before coming back out.
Referee Andres Cunha, though, simply checked the goal-line monitor on his watch and pointed to the centre circle as Australia keeper Mat Ryan stood appealing with the ball in his hands.
The Uruguayan official had earlier stood by the side of the pitch to watch the challenge by Josh Risdon on Griezmann on the Video Assistant Referee (VAR) monitor and decided the Australia right back had brought the French striker down in the area.
CONTROVERSIAL TECHNOLOGY
Technology was never going to rid football of controversy, however, and Australia coach Bert van Marwijk was not alone in thinking that it was not a foul with Risdon making contact with the ball before Griezmann tumbled over his trailing foot.
There was no doubting the quality of the approach play to Pogba's goal, though, after he exchanged one-twos with two team mates to carve his way through the middle of the Australia defence.
It was a rare moment of fluency from the French, however, with the much vaunted attack of Griezmann, Kylian Mbappe and Ousmane Dembele unable to create opportunities when pace alone was not enough to get past the wall of Australian defenders.
The trio did do enough to ensure France had the better of the opening half but Australia had the best chance to score before the break when a deflection off Frenchman Corentin Tolisso from an Aaron Mooy free kick was well saved by Lloris.
Deschamps's France side will move on to their remaining Group C clashes against Denmark and Peru, who meet later on Saturday, with three points but plenty to do to justify their place among the pre-tournament favourites.
Australia and their 10,000 gold-shirted fans will be disappointed that all their hard work counted for nothing as they seek to get out of the group stage for the second time in their fifth World Cup.
"I'm proud and disappointed, I think we gave a very good performance, nobody had expected that we could play like this against France," said van Marwijk.
"I think there were a lot of moments they didn't know what to do any more... a lucky goal just before the end... we deserved a draw."
(Editing by Ken Ferris)
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