Begu claimed the title at The Tashkent Open in Uzbekistan. She lost just one set in a run that included upsets of three seeded players, including No. 2-seed Urszula Radwanska. Begu jumps from 87th to No. 59 in the rankings.
Flipkens beat seeded players in the semifinals and finals in taking the championship at the Challenge Bell in Quebec City. Her 6-1, 7-5 decision over eighth-seeded Lucie Hradecka in Sunday's final moves Flipkens from 102 to No. 69 in the rankings.
The Top 10 is unchanged this week with Victoria Azarenka, who has 10,265 rankings points, leading the way for an 11th consecutive week and 30th week overall.
Angelique Kerber, who has 5,085 points, is sixth and Sara Errani comes in at No. 7 with 4,755. Li Na has 4,526 points for eighth place and Samantha Stosur (4,200) and Marion Bartoli (3,800) complete the Top 10.
Bartoli, the top seed at the Guangzhou International Women's Open, is the only Top 10 player in action this week. No. 11-ranked Carolina Wozniacki is seeded first at the KDB Korea Open in Seoul.
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'
Benedict Cumberbatch's dramatic reading of R. Kelly lyrics is just what you need