The Gunners roared into an early lead as Swedish forward Stina Blackstenius, who scored the only goal of last season's final against Barcelona, headed the opener from Katie McCabe's free kick in the 22nd minute.
Arsenal went in 2-0 up at the break thanks to a superb long-range strike from England winger Chloe Kelly, but Chelsea's Lauren James lashed an even better effort into the top corner in the 66th minute to reduce the deficit.
With the wind howling around the Emirates Stadium and the rain pouring down, Russo restored Arsenal's two-goal cushion ahead of the second leg as she latched on to a Blackstenius pass and steered the ball past Hannah Hampton.
Earlier in the evening, a deflected Lineth Beerensteyn shot gave German side VfL Wolfsburg a 1-0 win over eight-time champions OL Lyonnes ahead of next week's second leg in France.