The American is the fourth seed at this event and with the top three players all withdrawing, was a huge fan favourite against the ninth-seeded Russian.
"I've never played in front of a crowd that's really, really been so behind me," Williams said.
The partisan support had a lot to be happy about as did Williams, who won four straight games in the second set to close out the match.
In other quarter-final action, 11th seed Patty Schnyder caused the biggest upset of the round when the Swiss left-hander defeated fifth-seeded American Lindsay Davenport 6-3, 6-2.
Schnyder needed just 52
Davenport lost the first four games of the first set and never seemed to recover.
In the semi-finals, Williams plays Croatia's Jelena Kostanic, who defeated Hungary's Petra Mandula 7-6, 6-2 to stay on course for her first tour victory.
Schnyder meets 16th-seeded Conchita Martinez of Spain in Saturday's other semi-final.
Martinez celebrated her 32nd birthday in style, easing past eighth-seeded Nadia Petrova of Russia 6-3, 6-1.