Austin's brace sealed a 3-1 win for Saints and took him to an impressive tally of seven goals in his past six games.
But Puel is still refusing to guarantee Austin a starting place on Thursday when Southampton visit Inter Milan in the Europa League.
"He was a little ill this week and I didn't know if he could play," admitted Puel.
"It is important for the team with the character and a good spirit of course, but all the players can make a good game around him.
"You saw a very good player and a very good team in the second half [against Burnley].
"Nathan [Redmond] also came in with a very good performance, with very good technique. I asked him to shoot a bit more when he gets the chance and he did that."
Puel, who likes to rotate his forward players, could change his policy and pair Austin with Redmond again at the San Siro.
But, when asked if Austin had made himself undroppable, the Frenchman responded: "No, it's important he doesn't play all the games from the start.
"It's important you don't lose Charlie, it's important he's available every time, to keep a good physical level for every player.
"Charlie is not an endurance player; it's important to keep him with his qualities, and we can change the team every game. It's not possible, for me, for a player to start all the games every three days.
"We can keep all the players with a good performance all the time and a good physical level. It's important to conserve (their energy), to continue this work for the team, for the future."