After restricting Hobart to 153/7, with Mohammad Rubel taking three wickets and Ripon Mondal claiming two, Bangladesh HP endured a shaky start in the chase.
Habibur Rahman Sohan, Ariful Islam, Jishan Alam and captain Irfan Shukkur fell cheaply, while Shamim Hossain Patowary made 16.
But Sabbir anchored the innings with a fluent 59 off 45 balls, including six fours and two sixes, before falling soon after reaching his half-century.
Abu Haider Rony and Abdul Gaffar Saklain then guided Bangladesh home with three wickets to spare.
Earlier, Tig Willey remained unbeaten on 78 off 53 balls to lift Hobart to 153/7 after they struggled early. Rubel struck three times and Ripon took two wickets to keep the hosts in check.
The victory gives Irfan Shukkur's Bangladesh HP back-to-back wins after their success against Nepal.
