Pool stage victories over Georgia, Australia, Fiji and Uruguay plus the most nail-biting of quarter-final successes against France have sent Wales into the last four.
In Oita, Biggar played a key role as Les Bleus – who had lock Sebastien Vahaamahina sent off – were beaten 20-19.
He said: “We have to count ourselves a bit lucky, but if you want my honest opinion I couldn’t care less.
“We are in a World Cup semi-final and have played some good rugby in this tournament so far. A lot of teams wouldn’t have the character and desire we have to find a way. That’s what’s important.
“You get no points for style and no bonus points at this stage of the tournament. The coaches know they are lucky they have a squad of players here who will dig as deep as anyone.
“Looking at the quality of the rugby in the other quarter-finals we were certainly below that, but what we have in abundance and as much as anybody is a desire and a fight not to give in.
“That’s something that can’t be trained. To play South Africa we are going to have to be a lot better – there is no getting away from that. We are very relieved, but also happy.
“We don’t do that very well in Wales! We have managed to reach a semi-final of a World Cup and we are frustrated with the way we’ve played.”
Biggar kicked two conversions and two penalties against France and produced a controlling display from fly-half. Still, Wales know they can play much better than they did last time out.
“The red card was a massive turning point, let’s not hide behind that,” said Northampton’s Biggar.
“When you get to a semi-final you probably think that on their day any one of four teams can win in it then. We watched the All Blacks and they were pretty exceptional.
“But we are happy to keep under the radar and go about our work quietly. There certainly won’t be any lack of motivation as we are 80 minutes from getting into the World Cup final.”
Wales have won all of their last four meetings with South Africa, but their last defeat to the Boks was at the 2015 World Cup’s quarter-final stage. Rassie Erasmus’ current crop are a very efficient outfit.
Biggar and Wales are determined to seal global glory for Gatland as the World Cup will be his last tournament in charge.
The New Zealander has won the lot in the northern hemisphere since taking charge in 2008, but there is one trophy which still eludes him.
Biggar – who will be joined in Japan by wife Alex and son James this week – confirmed he is fighting fit after suffering two consecutive head knocks in the pool stage wins over Australia and Fiji.
The 30-year-old said: “I’m massively looking forward to South Africa. It has cost me a few quid as I’m flying the wife and little one over now – that’s the only downside of it! It’ll be great to have them out here. Ultimately that’s the most important thing and why a lot of us are doing this – the sacrifices they are making. It’s a few quid down the drain, but that’s OK.
“I think my wife actually booked the tickets at half time in the France game with a cancellation policy. It wasn’t looking great at half time! We have two games to potentially change the rest of our lives. It’s really exciting.
“We are all motivated for ourselves. We don’t want to get caught up in it being Gats’ last one or two weeks or whatever. Hopefully we can deliver a much better performance next week, but ultimately we’d take the same performance if we come out on the right side of the scoreboard as it’s all about winning rugby games now.
“If we do two in two weeks it would be pretty special.”