Johnson finished with the stunning figures of 2-6 from four overs as Australia restricted the Tigers to a paltry 8-109 from their 20 overs in the third and final match at Chattogram on Sunday.
The tourists cruised to victory with 54 deliveries to spare, finishing on 3-112.
Captain Mitch Marsh made light work of the modest target, bludgeoning 60 off 28 balls - including seven fours and four sixes - to be named player of the match.
"I'm not sure how I got man of the match - I think Spencer Johnson took 2-6 off four overs, so maybe he should be up here," Marsh said sheepishly at the post-match presentation.
"The way we adapted to the conditions each and every game and took our learnings, I thought we were fantastic with the ball today."
Johnson set the tone, conceding one run off the second over of the innings, one off the fourth, two off the ninth and, most impressively, just two off the 20th.
His analysis included a wide and a no-ball, meaning only four were scored off the bat.
Experienced duo Nathan Ellis (2-21) and Adam Zampa (2-22) provided valuable support for Johnson, who bounced back superbly after conceding 39 from two overs in the second game on Friday.
Acting skipper Towhid Hridoy (61 not out) waged a solo vigil for the hosts, with Rishad Hossain (16) the only other Bangladeshi batter to pass seven.
After losing the ODI series 2-1, it was a resounding fightback by Australia in the 20-over format.
Their win appeared a formality from the moment the Tigers crashed to 3-11 after five overs.
The procession started when opener Tanzid Hasan (5) was caught short at the non-striker's end as Saif Hassan's straight drive deflected off Johnson and onto the stumps.
Saif (1) and Pervez Hossain Emon (1) holed out tamely after struggling badly for timing, chewing up 22 deliveries between them.
Zampa captured his 350th international wicket when he nutmegged Nurul Hasan (6) with a slider, before Bangladesh slumped to 5-36 when Shamim Hossain (0) gloved Johnson to wicketkeeper Josh Inglis.
The Aussies were untroubled in their reply.
Inglis (17) holed out before opening partner Marsh did likewise, soon after raising his 14th T20 half-century.
Cooper Connolly (15) miscued leg-spinner Rishad to Saif, before Tim David entered with eight required and calmly clubbed two sixes.
