The Scottish champions will now play Ukrainian champions FC Zhilstroy at the same venue on Thursday for a place in the last 32 of the Uefa Women's Champions League. The two sides have maximum points in Group 4 after Zhilstroy beat Nove Zamky 3-1 yesterday.
City, showing eight changes from the team which beat Nove Zamky on Saturday, quickly established their superiority. Nicola Docherty and Carla Boyce combined down the right to set up a chance for Susan Fairlie after just 90 seconds, but the striker screwed the ball just wide of a post.
In a totally one-sided first half, the Glentoran goalkeeper Ashleigh Mckinnon had a number of shaky moments in the difficult conditions, but luck and poor finishing kept the sides level until four minutes from the interval, when central defender Courtney Whyte sent a low shot past Mckinnon.
City lost their captain, Leanne Ross, to injury early in the second half but brought on Irish internationalist Denise O'Sullivan and tightened their grip even further.
Leanne Crichton had a shot tipped on to the post by Mckinnon, just one of several close calls as the home side rained in shots on the Glentoran goal. In the space of 60 seconds Crichton had a shot tipped over the bar and another effort saved on the line.
The failure to add to their tally could present problems for City against Zhilstroy, who have a better goal difference, and a draw would be good enough to take the Ukrainians into the knockout stages.