Makarova, the 22nd seed, had never been past the third round in her six previous appearances at Wimbledon.
But the 26-year-old, a quarter-finalist at the Australian and French Opens last year, destroyed the world number four 6-3 6-0 and will play Czech 23rd seed Lucie Safarova for a place in the semi-finals.
Radwanska's exit means three of the top four seeds have been eliminated from the women's draw following the defeats of Williams and Li Na.
Earlier, Zahlavova Strycova moved into the last eight with a 6-2 7-5 victory against former world number one Caroline Wozniacki.
Zahlavova Strycova, the world number 43, had enjoyed the best win of her career in the previous round when she defeated Chinese second seed Li and she added Danish 16th seed Wozniacki to her list of upsets.
The 28-year-old, who served a six-month ban for doping last year, will play compatriot Petra Kvitova for a place in the semi-finals.
Sixth seed Kvitova, who won the Wimbledon title in 2011, routed world number 61 Peng Shuai of China 6-3 6-2 in just one hour.