The match, held at the Jamtkraft Arena, saw Syria international Daleho Irandust open the scoring in the 12th minute, setting the tone for Brommapojkarnaâs dominant performance.
Botchway, an 18-year-old Ghanaian who joined from Kowa Naso before his stint with Lions, extended the lead just before half-time.
His first goal came after a deft piece of play, and he struck again shortly after the interval with another well-placed shot.
Icelandic defender Hlynur Freyr Karlsson then added to the tally in the 62nd minute with a powerful finish, while midfielder Ludvig Fritzson capped off the scoring moments before the final whistle.
Since his move to Sweden in April, Botchway has now netted four goals across all competitions, showcasing his growing impact on the pitch.