The fifth seeded Hungary-Kazakh pair proved too good, breaking twice in each set for a place in the semi-finals.
Babos and Shvedova broke in the third and seventh game of the first set.
In the second set too they effected breaks twice, in the opening and sixth games, and looked on course for a comfortable victory before the Indo-Swiss pair hit back with a break in the eighth game to make the score 3-6.
However any hopes of a fightback were dashed as the fifth seeds served out the next game to clinch victory in 68 minutes.
Sania had already exited from the the mixed doubles. She and her Croatian partner Ivan Dodig lost in the second round to unseeded the British duo of Neal Skupski and Anna Smith 6-4, 3-6, 5-7 in a match that lasted over two hours.
IMAGE: India's Sania Mirza, right, and Switzerland's Martina Hingis
Photograph: Shaun Botterill/Getty Images