The undefeated students finish top of the north-east group with seven points from their three group fixtures and qualify for the last eight alongside runners-up Bo'ness United.
David Maskrey and Patrick Baird netted the all-important goals for Dorian Ogunro's men with Taylor Steven on target for visitors Strollers at East Peffermill.
However, there was disappointment for Spartans who finish bottom of the section following their 3-1 defeat to Bo'ness at Ainslie Park.
Gregor Nicol gave the away side a first-half lead only for Kevin Waugh to equalise for Dougie Samuel's men nine minutes after the half-time interval.
Former Hibs winger Danny Galbraith restored Bo'ness' lead before Nicol grabbed his second and the visitors' third to seal the victory.
Edinburgh Uni will learn who their quarter-final opponents are in due course.
They are joined in the last eight by Broomhill, East Stirlingshire, Berwick Rangers, East Kilbride and Caledonian Braves.
There will be a playoff between Gala Fairydean Rovers and Vale of Leithen at a neutral venue after the Borders teams could not be separated.
Lowland League champions Bonnyrigg Rose pulled out of the League Cup to concentrate on the pyramid play-offs.
Robbie Horn’s men were not in action at the weekend but travel to Rangers B on league duty on Tuesday night.