They will now face each other next Sunday in Yokohama, a day after England play New Zealand at the same venue for a place in the final.
Here, the PA news agency wraps up Sunday's action.
Wales battle back
Wales fought back from 19-10 down at half-time to win 20-19 in Oita against the French, whose cause was hampered by the dismissal of Sebastien Vahaamahina for an elbow on the head of Aaron Wainwright shortly after the break.
Dan Biggar subsequently reduced the deficit with a penalty before Ross Moriarty's 74th-minute try, converted by Biggar, sealed victory for the Six Nations Grand Slam champions.
Head coach Gatland said afterwards: "I'm very proud of our players because they never give up even when they're under a bit of pressure. They keep fighting and finding a way to get a result.
"The red card was significant but that sometimes galvanises teams as well. We didn't play our best but we showed great character and that's testament to this group of men and now we can look forward to the semi-final."
South Africa advance
Wales now take on a Springboks outfit who saw off Japan 26-3 in Tokyo.
After South Africa took the lead through Makazole Mapimpi's try, they had Tendai Mtawarira sin-binned for a crude tackle midway through the first-half.
Japan's response of a penalty saw the score at 5-3 at the interval, and they were then overpowered in the second half as Faf De Klerk crossed and Mapimpi added his second try.
While Wales are a game away from a maiden Rugby Word Cup final, South Africa are looking to reach their first since 2007.
Fond farewell to the Brave Blossoms
Japan, who had never played in the quarter-finals before this, exit the tournament having made a major impression and won many admirers.
Adding to their famous shock victory over South Africa at the 2015 edition, the Brave Blossoms pulled off eye-catching victories over Ireland and Scotland en route to topping Pool A.
Jamie Joseph's side have been eliminated but can certainly hold their heads high about what they have achieved on home soil.
Cheika to step down
A day on from Australia's 40-16 defeat to England in Oita, Michael Cheika announced he is to step down as Wallabies boss.
The 52-year-old said he had to resign after pledging to the Australian Rugby Union that he would if the side fell short of their potential in Japan.
Cheika had been in charge since 2014 and saw his team finish as runners-up at the 2015 World Cup.
Yielding 29 points, South Africa's win against Japan was the lowest scoring match at the tournament so far.