The win took Mahmudullah’s men to the top of the points table with five wins in six matches, while Rangpur slumped to their third defeat in seven matches on Thursday.
The Riders had put up an ordinary show with the bat, posting 104 for 9 before rain held up play for about an hour during the chase with the Bulls on 25 for 1 in 3.3 overs.
The target was then revised to 75 in 13 overs as Mahmudullah led his team to the win with an unbeaten 23 with three balls to spare.
They lost Evin Lewis (4), who scored a century in the last match, to a leg-before decision early on before drizzle began at the Zahur Ahmed Chowdhury Stadium in Chittagong.
After play resumed, the spinners removed Rony Talukder (23) and Mehedi Maruf (3) to give their team a glimmer of hope but Mahmudullah and Nadif Chowdhury (16) steadied the ship and settled in to snuff out any chance of a Riders comeback.
Chowdhury departed in the penultimate over with his team three runs shy of the win before the skipper calmly finished off the chase with a single.
The Bulls had opted to leave out their best bowler, Devon Cooper, but he was hardly missed as Al-Amin Hossain, Mohammad Sami and Rayad Emrit kept the batsmen on a leash while regularly taking wickets.
But the Riders had only their batsmen to blame for the below-par total.
They kept charging down the track and, in the process, threw their wickets away before the lower order managed to haul them past 100.
Soumya Sarkar went back for a golden duck when he tried to slash hard at a short and wide delivery, sending a tame catch to cover to give Al-Amin his first scalp that commenced the slide.
Shakib (1), Mohammad Mithun (2), and Thisara Perera (9) all went back off mishits to leave the Riders five down on 54 in the 13th over.
Jahurul Islam (24) had watched his partners fall at the other end but then joined the act when he miscued an Emrit slower to cover.
They still had hopes of a good score with Misbah-ul-Haq (22) in the middle. He executed two impeccable reverse sweeps for two fours before Sami banged one in short that brushed Misbah’s gloves on its way to the keeper.
Wahab Riaz (12 off 10) and Al-Amin Chowdhury (17 off 13) then cracked four boundaries and a six between them to help their team to the total.