Azarenka secured a spot in the semifinals and next will play Maria Sharapova.
"It's kind of difficult to believe that a little girl from Belarus is on that list," said Azarenka. "It's really incredible achievement. When I started to play tennis I had this big picture in my head that I want to be there. Back then it's so far away. It's like pretty much touching the sky.
"I'm here sitting kind of in the sky, so it's really incredible feeling."
Sharapova swept past Samantha Stosur 6-0, 6-3 on the final day of round-robin play, but Azarenka's win means that the Russian will not be able to wrest the No. 1 ranking form her.
Azarenka won the Australian Open in January and had been ranked No. 1 for most of the year, except for a brief period after Sharapova won the French Open.
Sharapova had already qualified for the semifinals of the elite, eight-woman event and has won all three of her round-robin matches.
The Russian rolled through the first set, allowing Stosur only five points. Stosur was an alternate who replaced defending champion Petra Kvitova, who withdrew because of illness.
"I certainly didn't want to play as long a match as I did in the previous round. I wanted to do a better job of a few things and I think I did," said the second-ranked Russian.
Stosur put up more of a fight in the second set, but Sharapova ended it with a smash.
Errani finished the tournament with one win and two defeats. The seventh-seeded Italian won a game in the middle set that went to nearly 15 minutes but was not able to use that momentum and dropped her serve to fall behind 6-5.
Radwanska served out the set and opened up a decisive 3-0 lead in the final set to secure a place in the semifinals.
"I had to play my best tennis until the end of the match because every game was a big fight," Radwanska said.
Williams was undefeated in three matches in her Red Group and didn't play on Friday.
The match between Radwanska and Errani lasted 3 hours, 29 minutes and was the longest recorded best-of-three-set match in WTA Championships history.
The previous record was 3:24, set in the 2007 final when Justine Henin beat Sharapova 5-7, 7-5, 6-3. The event, however, also used a best-of-five format for a while and had longer matches.
"Altogether I am exhausted," said Radwanska, who had another three-hour-plus match in the previous round. "I really need to be ready for tomorrow."