It looked like City's hopes of a comeback had been extinguished after just three minutes when, as she did in the first leg, Mille Gejl opened the scoring for Koge.
The former Crystal Palace midfielder took a short corner to Rikke Madsen and latched on to her fellow Denmark international's cutback from the byeline before steering her low drive into the far corner from just inside the penalty box.
However, City were pinging the ball about far quicker than in the first leg in Denmark and were back level on the night within six minutes.
Captain Amy Muir set Nicole Kozlova down the left wing and the former Koge player's low ball into the penalty box was fired home by strike partner Abby Harrison.
A couple of fine saves from goalkeeper Lee Gibson denied the visitors before City were back level on aggregate on 25 minutes when Emily Whelan headed powerfully home after left-back Muir crossed to the back post.
City's joy only lasted two minutes as veteran former Manchester City and Denmark forward Nadia Nadim turned in from the left wing and sent a right-foot drive looping over Gibson and into the far corner.
Glasgow were ahead again on the night when Harrison rose with the visiting goalkeeper to meet a Lisa Evans corner and the ball deflected off Emma Pelkowski and into her own net.
Fellow midfielder Sarah Thygesen pounced on an Emma Brownlie error to slot what looked like a killer strike for the Danes six minutes from normal time.
However, Brownlie made amends by rising to meet another Evans corner and level the score on aggregate as the game entered added time.
Just when it looked like extra time might be needed, Lisa Forrest headed a cross from fellow substitute forward Sofia Maatta over the stranded goalkeeper to signal joyous scenes among the home support.
