Lauren James pulled a goal back for Chelsea to keep the Blues in the tie but reigning European champions Arsenal will still be well-placed come kick-off in next week's second leg at Stamford Bridge.
The Women's Champions League was the one major title that eluded Chelsea during Emma Hayes' trophy-laden 12-year reign and the Blues are still seeking to be crowned queens of Europe for the first time since Sonia Bompastor replaced the now-United States boss in 2024
Chelsea were twice denied by the woodwork early in Tuesday's tie.
After just five minutes, Alyssa Thompson's cross looped off the boot of Laia Codina and spun over goalkeeper Anneke Borbe before rebounding off the far post.
Two minutes later, James saw a shot hit the same post before the ball fell clear.
Arsenal, however, went ahead in the 22nd minute when Sweden striker Blackstenius headed home her first Champions League goal of the season from a Katie McCabe free-kick.
The Gunners doubled their lead 10 minutes later when Beth Mead played a crossfield pass to Kelly, who took a touch before lashing a 20-yard drive underneath another England international in Blues goalkeeper Hannah Hampton.
Chelsea had the ball in the net five minutes before half-time but Veerle Buurman's effort was disallowed for a shove on Laia Codina - a borderline decision upheld by VAR to the obvious annoyance of Bompastor.
Chelsea gave themselves a lifeline when James left fly in the 66th minute with a brilliant left-footed curling lob from some 25 yards out.
But Russo restored Arsenal's two-goal advantage 14 minutes from time when, after being played in by Blackstenius, she flicked the ball up and volleyed past Hampton in fine style.
Chelsea suffered fresh frustration in stoppage time when Kadeisha Buchanan bundled the ball into the net only to be penalised for a foul on Borbe.
The sides meet again on Thursday April 2 with the winner set to meet Wolfsburg or eight-times champions Lyon with the German side leading 1-0 after the first leg.
Check out the full match statistics on Flashscore.
