Biggar kicked 23 of Wales's points in last weekend's 28-25 win that helped send hosts England crashing out.
But the 37-times capped Ospreys player said: "Probably off the field I am the least confident person there is.
"When I am on the field it is when I am most comfortable and almost when I can relax a bit more in terms of knowing what I'm doing."
The 25-year-old added: "I don't like all the publicity and everything that goes with doing well, but I know it is part and parcel of it."
If Wales beat Australia in Saturday's Pool A decider at Twickenham they will snap a 10-match losing streak against the Wallabies stretching back to 2008 - but more importantly secure a quarter-final against either Scotland or Japan.
Defeat would send Wales into a game against two-time world champions South Africa on 17 October.
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"Everyone is talking up the Wallabies this week, and quite rightly so as they are an outstanding team," said Biggar.
"Their backs are looking sharp, and they played some really good rugby against England, so it is going to take a heck of a defensive effort from us.
"But we went to Twickenham and beat England in their own World Cup, so we have got to go in there with plenty of confidence and back ourselves to come out on top.
"We go into this weekend with a fraction of pressure off us - or as little pressure off as you are going to get in an international World Cup rugby match - but knowing that we have qualified no matter what has put a relaxed mood in the camp."
Wales fly-half Dan Biggar says he does not enjoy the increased attention his World Cup success has brought him.