It took until the 54th minute of the match at Skytteholms IP before AIK defender Oskarsson, with a well-placed shot from wide, was able to break the deadlock against Brommapojkarna.
The Brommapojkarna tried to push for an equalizer, but never managed to produce a proper final push.
The victory means that AIK now sits in third place in the table, two points ahead of fourth-placed Malmö. BP is in ninth.
Friday's second women's Swedish football match saw Växjö win 2-0 against Rosengård.
