Left-arm spinner Shakib Al Hasan grabbed three wickets for eight runs as Afghanistan, sent in to bat, were skittled for 72 in 17.1 overs at the Sher-e-Bangla stadium.
The meagre total was the second lowest in the World T20 competitions, barely surpassing Ireland's 68 against the West Indies in 2010.
A sell-out home crowd of 25,000 then cheered every run as Bangladesh raced home in 12 overs, with opener Anamul Haque sealing the emphatic win with a six off Samiullah Shenwari.
Haque, who put on 45 for the first wicket with Tamim Iqbal, was unbeaten on a 33-ball 44 that was studded with three sixes and four boundaries.
With just one team going into the Super-10 stage from the preliminary group A that also has debutants Hong Kong and Nepal, the defeat left Afghanistan facing an early exit.
Shakib, who was named man of the match, said the turning pitch helped his team make a good start to their campaign.
"There was obviously spin out there," he said. "We put the ball in the right areas and that did the trick for us. This is usually a good track, but today it was different."
The Afghans, who had beaten Bangladesh in the preceding Asia Cup one-day tournament, faltered badly on Sunday to reduce a much-anticipated contest to a one-sided affair.
Skipper Mohammad Nabi admitted Bangladesh's spinners did not let his side settle down after hard-hitting opener Mohammad Shahzad had been dismissed off the first ball of the match.
"We were under pressure in this opening match," said Nabi. "Shakib and the others bowled very well on the spinning pitch. But I am hopeful we will put up a better performance in the remaining matches."
Six qualifiers -- Ireland, Afghanistan, Nepal, Hong Kong, the United Arab Emirates and the Netherlands -- joined Zimbabwe and Bangladesh in the preliminary league of two groups.
The two group winners will then contest the Super-10 round with the top eight Test nations -- Australia, Pakistan, South Africa, England, New Zealand, India, Sri Lanka and the West Indies -- from March 21.
The final is on April 6 in Dhaka.
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