After a goalless 90 minutes, the Sweden international finished off a smart move to wrap up a deserved 1-0 win at Meadow Park.
Jonas Eidevall's side will now face either West Ham or Chelsea in the final at Selhurst Park on March 5.
Laura Wienroither missed a one-on-one as Sandy MacIver made a smart save, while Women's Euros heroine Chloe Kelly was denied by Sabrina D'Angelo at the other end.
The closest either side came to breaking the deadlock in regulation time was Lina Hurtig hitting the crossbar from close range.
Arsenal built up a head of steam in the closing stages of normal time but a mixture of profligate finishing - especially a poor header from Hurting - and a couple of smart saves took the game into extra time.
The hosts finally found a breakthrough just three minutes into the extra period as a Hurtig cross was bundled home by Blackstenius, who has seen the goalscoring onus fall on her shoulders following the serious knee injuries to both Beth Mead and Vivianne Miedema.
Arsenal will now be hoping for a similar result when the two teams meet again in the Women's Super League on Saturday morning.