Biggar was forced off the pitch in the 12th minute of the 40-6 win at OL Stadium which resulted in Wales becoming the first team to reach the quarter-finals, which start on October 14.
"He should be a couple of weeks, no more than that," forwards coach Jonathan Humphreys told reporters on Monday.
"He's feeling a lot better this morning."
Wales fans will take some solace from the fact that backup flyhalf Gareth Anscombe was Man of the Match against Australia, making one try as well as kicking six penalties, a conversion and a drop goal.
"He's a huge professional. So well prepared," Humphreys added.
"We've got a couple of leaders in the group, Bigs and Gareth are two of those people. They drive how we want to play, they drive training standards. He was excellent."
Humphreys said the players would not quickly forget the passionate support they received from the massed ranks of their red-shirted compatriots in Lyon.
"Definitely a memory to cherish," he said. "You can feel it as players when you see the middle of cities just full of red, people singing the national anthem. It's awesome."
Wales still have one more Pool C match in Nantes on October 7 against Georgia, who stunned Wales 13-12 in Cardiff last November.
"There are no wrongs to right. They deserved it, they were excellent on the day," said Humphreys.
"This is a totally (new) thing here. This is not about revenge or anything like that, it's about making sure we're at our best to win the game and top our group. That's it."