The fly-half went off in the 28th minute of a pulsating Pool D victory over Australia two days ago after making a try-saving tackle on Wallabies centre Samu Kerevi.
He failed a head injury assessment, and was replaced by Rhys Patchell, who kicked 14 points as Wales clinched a memorable 29-25 win.
"Dan is symptom-free at the minute, and obviously just doing his protocols as you normally would," Wales assistant coach Neil Jenkins said.
"As far as I understand, he will be ready to go by the time we play Fiji (on October 9), so that's obviously good news.
"It's bumps and bruises other than that, normal stuff from a Test match of that size.
"Dan is an outstanding rugby player, a world-class rugby player, so he's a huge part of our team and the way we like to play the game.
"I think we must not forget how good Rhys Patchell was when he came on, and how good a player he is as well. We are quite lucky to have the strength in depth that we have in our squad."
Patchell's composed performance helped Wales successfully navigate their way to victory as they held off a turbo-charged Wallabies fightback to take charge of Pool D.
Asked if it was the best form he had seen Patchell produce in a Wales shirt, Jenkins added: "In a game of that size, you could probably yes, but he was exceptional last summer in Argentina and the Six Nations a couple of years ago against Scotland.
"He was excellent on Sunday. He came on and played the game we wanted to play and put us in the right areas and got us the lead we maintained to win the game."
Biggar and Patchell both kicked drop-goals in the game, which was double Wales' tally from that scoring method between the 2015 World Cup and Japan 2019.
"It was our first time for 30-odd games that we've dropped a goal," Wales' record international points-scorer Jenkins said.
"I think against Japan was the last one when Sam Davies nailed one, and Dan's last one I think was in the World Cup four years ago against South Africa.
"Shaun (Wales defence specialist Shaun Edwards) is always on to me, to be honest.
"Every clip he sees of some big World Cup game, there is pretty much a drop-goal in it, and obviously it proved vital for us on Sunday.
"It is a skill in itself. If you can nail those three points and keep ticking the scoreboard over - three, six, nine - it makes a big difference, and it did for us on the weekend.
"On most occasions that we were in Australia's half in the first-half on Sunday, we managed to come away with points.
"I am sure as the tournament progresses, we will see a few more (drop-goals)."
Wales will complete their pool phase by tackling Fiji and then Uruguay, and they are firm favourites to win the group, setting up a potential quarter-final against France or Argentina.
"It is a fantastic achievement what we've done, but it is two out of two," Jenkins added. "We have another two tough games to go against Fiji and Uruguay.
"There is no way Fiji are going to lie down for us a week on Wednesday, no chance. It is going to be a very tough game. It could even be harder than on Sunday in a funny way.
"If you take anyone lightly in this World Cup, as we have already seen, you are going to come undone, plain and simple. There are no easy games in Test rugby any more."
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies