Kanu got Andree Jeglertz's side off to a bright start when she opened the scoring after just nine minutes of action.
However, the hosts got back into the mix when Jenny-Julia Danielsson netted the equaliser thanks to Honoka Hayashi's assist.
Despite AIK's bounce back, the visitors earned the winner two minutes from half-time thanks to Frida Leonhardsen Maanum's strike.
Nigeria's Kanu, who lasted the duration for Linkopings, has now scored once in two games and four times in all competitions this term.
Her teammate and compatriot Chinaza Uchendu continued to inch closer to full fitness since her unfortunate injury last June.