After winning the Swedish Cup earlier this year, Hammarby also secured a qualifying spot for UEFA's new tournament League League, the step during the Europa League.
Now the Stockholm team has stepped into the tournament, and they got a promising start to the European adventure.
After a goalless first half, it took ten minutes before Astrit Selmani, on his own, took the ball into the offensive third and put 1-0.
Just five minutes after the lead goal came Maribor's equalizer, but the guests' joy was short-lived.
Akinkunmi Amoo and Gustav Ludwigson combined and 24-year-old Selmani poked Bajen's second goal from close range.
The same Selmani also made it 3-1, after substituting talent Williot Swedberg collapsed in the penalty area and arranged a penalty.
The return in the Conference League qualifier is played on July 29.