It was in the ninth minute that AIK goalkeeper Milla-Maj Majasaari came out and kicked the ball straight to Linköping's Uchenna Grace Kanu.
Kanu lifted the ball towards goal and it went in via an AIK defender's head.
The equalizer came quickly when the Stockholmers' Jenny-Julia Danielsson nicely placed the ball.
Just before the break, visiting Linköping made it 2-1 and again it was via an unlucky bounce.
Frida Maanum shot from outside the penalty area and the ball went through AIK's left back Linda Hallin who tried to cover but who instead steered it behind Majasaari.
The victory was Linköping's first in this year's damallsvenska after they lost the home premiere against Rosengård with 1-0.