World number one Williams, whose sister Venus lost to Flipkens in round one having won the first set 6-0, needed just 56 minutes to sink the 13th-seeded Belgian 6-0 6-3 and book a last-eight meeting with Magdalena Rybarikova.
"After I won the first set I definitely thought about Venus, and I thought about what a good comeback Kirsten had, so I thought, 'I'm not going to get over-confident'," Williams said on www.wtatennis.com.
"She's obviously a really good player, so I was just trying to stay really focused as well as really intense at that moment."
Rybarikova progressed after Wimbledon champion Marion Bartoli retired early in the second set due to an abdominal injury after losing the first on a tie break, with the score at 7-6 (7/5) 1-0 in her Slovakian opponent's favour.
Third seed Agnieszka Radwanska came through her third-round meeting with American Sloane Stephens 6-1 7-6 (7/2) to earn a quarter-final place against fifth seed Sara Errani, who defeated Alize Cornet 7-5 7-6 (7/3).
Sixth seed Petra Kvitova, the 2011 Wimbledon champion, made light work of 2011 US Open winner Samantha Stosur, overcoming the Australian 12th seed 6-3 6-3.
The Czech will take on Sorana Cirstea in the last eight after the Romanian toppled 15th seed Jelena Jankovic 6-3 6-4.
Chinese fourth seed Li Na came from a set down to beat Ana Ivanovic 3-6 6-1 7-6 (7/5), setting up a quarter-final clash with Dominika Cibulkova after the Slovakian prevailed 6-3 7-6 (7/4) against 10th seed Roberta Vinci.