Knowing they had to chase the game, Chelsea - who had manager Sonia Bompastor sent off late on - started at a furious pace.
Yet Arsenal soon knocked them out of their rhythm, and the home side didn't manage a single shot on target in the first half as the visitors took their time over every dead ball.
Arsenal grew into the game in the second half, but all the drama happened late on as Chelsea threw everything forward in an effort to get back in the tie.
Arsenal substitute Stina Blackstenius headed home from a cross from her Swedish compatriot Smilla Holmberg in the 80th minute, but the goal was chalked off after a VAR review found that the striker had been just offside in the build-up.
Chelsea went back on the attack and defender Veerle Buurman struck the post with a late shot before midfielder Sjoeke Nuesken was denied by a brilliant save from Arsenal keeper Daphne van Domselaar, who tipped her header onto the inside of the far post before the ball bounced away to safety.
Arsenal's Beth Mead then smacked a stoppage-time shot off the far post before Nuesken finally netted, stealing in ahead of defender Lotte Wubben-Moy to sweep home a first-time shot in the 94th minute and set up a nerve-jangling finale.
With the game reaching its climax, Arsenal's Katie McCabe wasn't penalised for pulling Alyssa Thompson's hair during a counter-attack, prompting an angry outburst that saw Bompastor sent from the dugout after picking up two yellow cards for her protests.
In the evening's early game, German side Bayern Munich scored twice late on to eke out a 2-1 win on the night and a 5-3 aggregate victory over Manchester United to qualify for the semi-finals for the first time since 2021.
Arsenal will face either Lyon or Wolfsburg in the semi-finals, with Bayern Munich set to take on Barcelona or Real Madrid, with Barca leading 6-2 on aggregate and Wolfsburg holding a slim 1-0 advantage heading into Thursday's second legs.