The Swedes had the better strikers, and that proved to be the difference between the teams.
Scott Booth made one change from the first leg, bringing Sarah Crilly in to play off striker Clare Shine instead of American import Haley Rosen.
City came unstuck after just seven minutes. Eskilstuna danger player Olivia Schough had already shown she was in the mood when she cut in from the right and beat two defenders before shooting weakly at Gemma Fay in the opening 60 seconds. This time she was on the end of good work by Malin Diaz and Annica Svensson to squeeze a shot home at the front post.
The goal left City needing at least three to go through to the last 16. The home side recovered from their shaky start and Eskilstuna defender Nathalie Bjorn was booked for one foul too many on Fiona Brown.
A couple of dangerous free-kicks into the box by captain Leanne Ross were as good as it got for City before teenage midfielder Erin Cuthbert showed the way with a powerful shot which just missed the target.
Booth’s side should have equalised in 37 minutes. Nicky Docherty swung over a great cross from the left but Shine missed the target from eight yards out.
City needed to find something in the second half and they scored in the opening minute. Fiona Brown sent over a cross from the left and Crilly prodded the ball home from close range. Fay kept City in the tie with a great save at the feet of Schough, but City had the bit between their teeth and Shine came close with a header, which Emelie Lundberg saved.
Just as it looked as if City might get themselves back into the tie, they were caught out by a second Eskilstuna goal on the break. Fay looked to have collected a low shot from Diaz, but she spilled the ball and Schough pounced to score her second.
With City now needing another three goals against the powerful Swedish side the tie was effectively over. The home side came very close twice in the last eight minutes, but powerful shots from Ross and Cuthbert just missed the target.