The 20-year-old striker scored four goals in eight games for BP last season. In June, Ölvestad suffered a cruciate ligament injury and is currently in rehabilitation.
"It's clear that it's been a tough time. I've tried to turn this time into something positive," Ölvestad said in a statement.
The new Häcken player's father is former professional hockey player Jimmie Ölvestad, who played for many years for Djurgården and in the NHL.
There are also family ties to football legend Lennart "Nacka" Skoglund.
"That's my great-grandfather. I understand that he is a major football icon in Sweden but also in Italy. Sometimes you hear that there are some similarities in the playing style," says Tuva Ölvestad.
The contract with the reigning champions is for four years.
