City's hopes of a comeback seemed to fade after just three minutes when, as in the first leg, Mille Gayle opened the scoring for Koge.
The former Crystal Palace midfielder took a short corner to Rikke Madsen and latched on to her fellow Danish international's shot from the touchline before sending her low shot into the far corner from just inside the box.
However, City moved the ball much faster than in the first leg in Denmark and came back on level ground within six minutes that evening.
Captain Amy Muir sent Nicole Kozlova down the left flank and former Koge player's cross into the box was headed home by strike partner Abbie Harrison.
A pair of fine saves from goalkeeper Lee Gibson denied the visitors a chance before City came back level on aggregate in the 25th minute when Emily Whelan headed home powerfully after left-back Muir crossed at the back post.
City's joy lasted just two minutes when former Manchester City and Denmark striker Nadia Nadeem turned from the left flank and fired a right-footed shot over Gibson and into the far corner.
Glasgow were ahead again on the night when Harrison, along with the away goalkeeper, met Lisa Evans' corner and the ball deflected off Emma Pelkowski into her own net.
Fellow midfielder Sarah Thygesen took advantage of Emma Brownlee's mistake to deliver what looked like a killer strike for the Danes six minutes from normal time.
However, Brownlee made amends by rising to meet another Evans corner and equalize on aggregate as the game entered stoppage time.
Just when it looked like extra time might be needed, Lisa Forrest headed striker Sofia Maattu's cross past the stranded goalkeeper, signaling joyous scenes among the home support.
