The striker's brace helped the Saints to a 3-1 win and gave him an impressive total of seven goals in his past six games.
Puel has regardless refused to guarantee Austin a starting place on Thursday when Southampton visit Inter Milan in the Europa League.
His so-far successful partnership with fellow goalscorer Nathan Redmond, however, could yet tempt a manager whose preference is to rotate his team.
"He was a little ill this week and I didn't know if he could play," said Puel of 27-year-old Austin.
"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 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."
Asked if Austin, who also has four goals in four games against his former club Burnley, 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."
The result took Southampton up to eighth but left 14th-placed Burnley just a solitary point above the bottom three and their manager Sean Dyche warned his team of the need to improve their away form.
"At home we play with bravery, fervour and energy," he said
"That has to come from us away from home.
"We know where we are and we know we have to develop quickly
Our home form has been good and we've looked solid.
"Away we haven't found that feel and we need that inner bravery to play.
"We are playing within ourselves and we can't do that."
Source : PA
Source: PA