The 2015 Damallsvenskan runners-up will travel to Scotland next week with a narrow 1-0 advantage over Glasgow City, thanks to a second-half goal from Mimmi Larsson.
Glasgow City played a 4-1-3-2 formation with Fiona Brown and Clare Shine playing up front, and those two nearly linked up in the opening moments, but Brown’s cross was just too high for Shine.
Eskilstuna was also showing a threat, and a mix-up in City’s defence nearly allowed them in when the bounce deceived Savannah McCarthy, but Gemma Fay was out to claim the ball.
City though was having a good spell with Brown having a speculative effort go wide and then Erin Cuthbert broke through the middle, beating several challenges before shooting wide.
A long ball from Jansson found Larsson in the City box but Fay was there to collect the ball, and then soon after, Jansson had her own effort saved by Fay.
In the second half, a quick throw-in caught out Glasgow City as Eskilstuna took the lead. Ingrid Schjelderup crossed to Larsson who headed in from six yards out.
City had a half-chance to equalise through another Brown effort as she shot across goal, but the keeper Emelie Lundberg made a good save.
The home side’s Petra Andersson came close through a shot that took a large deflection which forced Fay into a super save. The same player had a shot from outside of the box, but this time, she shot over.
Late on, most of the chances City fell to Hayley Lauder. The best of these was when Fiona Brown picked her out with a cross, but Lauder was unable to generate the required header.
Despite trailing, City showed enough that they have enough to fight back into the tie.