Home player Izabela Krizaj's attempt to head the ball clear instead turned into a perfect cross that set Skoog up for a free kick against Carina Schlüter.
The 21-year-old was cool there.
She slipped past the goalkeeper and gave visiting MFF the lead in the 20th minute at Nö Arena in Austria. Sixty minutes later, substitute Vesna Milivokevic made it 2-0.
Hammarby also won the first leg in the second qualifying round 2-0, when Elin Sørum, 26, and 16-year-old Fanny Peterson decided the home match against Scottish Rangers.
The return matches will be played in a week, on September 30. Victory on aggregate secures a place in the quarterfinals.
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