Stosur dispatched Kan 6-2, 6-3, giving the Australians an insurmountable 3-0 lead in the best-of-five opening round series and their first berth in the semis in 21 years.
The Russians came into the event with a 3-0 all-time mark against Australia.
The Aussies now go on to face the winner of the Slovakia-Germany match in the April semifinals. Heading into Sunday, the Germans held a 2-0 advantage in that series.
2014: The Year in Fashion [PHOTOS]
LeBron James reveals his new Nike shoes