Celtic beat Aberdeen 7-0, with Suzanne Mulvey scoring a hat-trick in the second round's only all-top flight tie.
Last season's runners-up Hibernian put 11 past Boroughmuir and Jeanfield Swifts routed Ross County 18-0.
Rangers, Dundee United, Dunfermline, Stirling University, Hearts, Renfrew, Glasgow Girls and Granite City also enjoyed comfortable victories.
Spartans and Forfar Farmington progressed after their games went to extra-time, while Raith Rovers edged out Dee Ladies.
Experienced Scotland striker Ross scored the first of her nine goals for Premier League leaders City after two minutes.
It was 10-0 when East Kilbride lost goalkeeper Amy Hiddleton to injury, with an outfield player putting on the gloves and letting in another five before the half time interval.
Ross, who has 122 international caps, completed her scoring with the 25th goal of the day, with Sarah Crilly finishing off the rout for a hat-trick.
Erin Cuthbert and Brogan Hay also grabbed three goals apiece.
While City are experienced Champions League campaigners, this is East Kilbride's debut season in Division Two Central, where they have won once in eight games.
The victory is not a record for Scotland's dominant women's team; that came in 2010 with a 29-0 success against Kilmarnock.
Scottish Women's Cup, second round:
Dundee United 8-0 Stenhousemuir
Hearts 3-0 St Roch's
Stirling University 10-0 Westerlands
Buchan 3-5 Forfar Farmington (AET)
Celtic 7-0 Aberdeen
Dee Ladies 3-4 Raith Rovers
Dunfermline Athletic 7-0 Dumbarton United
Glasgow City 26-0 East Kilbride
Hibernian 11-0 Boroughmuir Thistle
Hutchison Vale 5-0 Edinburgh Caledonia
Jeanfield Swifts 18-0 Ross County
Motherwell 1-4 Glasgow Girls
Rangers Ladies 7-0 East Fife
Spartans 0-0 Hamilton Academical (4-3 pens)
Renfrew 5-0 Queens Park
Dee Vale 1-5 Granite City
Glasgow City skipper Leanne Ross scored nine as the Scottish Women's Cup holders thrashed East Kilbride 26-0.