The 31-year-old paceman took three wickets in the post lunch session and added another in the final session, finishing with 5-24 to rout Bangladesh for a paltry 167 after they were set a 245-run target.
Best dismissed Shahriar Nafees (23), Shakib Al Hasan (two) and Mushfiqur Rahim (16) in quick succession and then had Bangladesh's top scorer Mohammad Mahmudullah (29) to give his team a 1-0 lead in the two-Test series.
The second and final Test will be played in Khulna from November 21.
Bangladesh, seeking their second win at home and fourth overall in Test cricket, lost openers Tamim Iqbal (five) and Junaid Siddique (20) in a disappointing start.
Siddiqui became Best's first victim in the innings before lunch.
But worse was to come as Best caught Nafees miscuing a pull off his own bowling and in his next over had Bangladesh's best batsman Shakib Al Hasan caught behind.
Rahim looked good initially but Best returned for a second spell to trap the Bangladeshi skipper in front of the wicket to derail Bangladesh to 85-5.
Sixteen runs later, Veerasammy Permaul had Naeem Islam lbw after the first innings century maker had made only 26. Permaul also dismissed Nasir Hossain for 21 and Sohag Gazi (19) to finish with 3-32 on his debut.
It was another disappointing batting display by the home team which ended in their 64th defeat in 74 Tests since gaining Test status in 2000 -- a matter of great criticism in the cricketing world.
Earlier, they were put on course for a win by off-spinner Gazi whose 6-74 are the best figures by a Bangladeshi bowler on debut.
Gazi had Darren Sammy in the third over of the day after the tourists resumed at 244-6 to raise hopes of a Bangladeshi win.
He then took two wickets in one over to record his maiden five wicket haul in his debut Test, cleaning up Ravi Rampaul (five) and Best (nought) off successive deliveries.
Shivnarine Chanderpaul, who hit 203 not out, came out to bat at number eleven because of illness and although he averted a hat-trick he was trapped leg-before for just one.
Gazi became only the third Bangladeshi bowler behind Naimur Rahman and Elias Sunny to grab five or more wickets in his debut Test, but his 6-74 is the best among the trio.