Serena Williams would meet a fellow American seed, either number 15 Sloane Stephens or number 23 Jamie Hampton, in the fourth round. Stephens defeated Williams in the Australian Open quarter-finals back in January.
Radwanska has 12 career titles having won back-to-back tournaments to start the year. She beat Yanina Wickmayer 6-4, 6-4 in the Auckland final before destroying Cibulkova a week later.
The Russian was just a fortnight removed from the French Open doubles title which she won alongside Ekaterina Makarova at Roland Garros. The player who lost her first six WTA finals before finally winning in January, admitted that conditions on court made life difficult.
Hampton squandered five set points in the opening set to give the momentum to her opponent, But Wozniacki was unable to serve out the second set leading 5-4 after winning the first in a tiebreaker and came up short in the third.
Hampton was the first US woman since Jennifer Capriati a decade ago to play the semis at Devonshire park and is the first qualifier on the WTA to reach a final this season.
Australian Open finalist Li was competing at the pre-Wimbledon event for the fifth time and had never advanced past the second round.
Radwanska, who won the title in 2008, struggled throughout the match at the Wimbledon warm-up tournament.
Capriati was inducted into the International Tennis Hall of Fame last year, but her career was overshadowed by personal problems, including incidents of shoplifting and drugs.
Hampton left the court for treatment on her back, but despite being in obvious pain she held serve for the set, with Azarenka slamming her racquet in frustration.
The victory keeps Azarenka on course to meet former world number one Caroline Wozniacki in the quarter-finals, and potentially Serena Williams in the semis, as she defends her first Grand Slam title.
Former World No. 1 Jennifer Capriati, a three-time Grand Slam champion who also won Olympic gold in 1992 at Barcelona, has been elected to the International Tennis Hall of Fame.
Former winner Jelena Jankovic became the first seed to crash out of the Indian Wells tournament on Friday, losing in straight sets 6-4, 6-3 to world No. 99 Jamie Hampton.
India's Rohan Bopanna and Pakistan's Aisam-Ul-Haq Qureshi, dubbed the Indo-Pakistan Express, bowed out the doubles quarter-finals of the 462,675-euro AEGON International tennis.
Somdev Devvarman and Sania Mirza crashed out of their respective events but Rohan Bopanna stormed into the quarterfinals of the men's doubles category as India rounded off a mixed day in the Aegon International tennis tournament on Wednesday.