England's strategy to play four spinners paid dividends as the 2009 champions restricted Bangladesh to 97-7 on a slow track in Sharjah as they managed to defend their lowest total in T20 World Cups.
England had made 118-7, riding on a 40-ball 41 by opener Danni Wyatt-Hodge.
Middle-order batter Sobhana Mostary played a lone hand for Bangladesh, making 44 off 48 balls, including a huge six off Dean over deep square leg.
The result was a disappointment for Bangladesh, who won against Scotland in the tournament opener on Thursday for their first T20 World Cup win in a decade.
"We missed a good opportunity to beat a good side. Our bowlers did a good job but our batters disappointed," said Bangladesh captain Nigar Sultana.
"On this kind of a surface, we needed to have a good partnership. We did not have a good powerplay, we gave away wickets."
England, who chose to bat first, started off cautiously before Wyatt-Hodge and Maia Bouchier (23) opened up to play some memorable shots.
"It was tough out there, conditions were tough for batting and boundary-hitting. Wyatt-Hodge and Bouchier did well," said England skipper Heather Knight.
"We targeted 135-140, that is what we prepared for."
Bouchier hit two fours on the trot in the fifth over off Marufa Akter as England raced to 36 runs off the first five overs.
Rabeya Khan dropped Bouchier on 16 off Marufa at point but returned to pick up her wicket, with Nahida Akter taking a catch at mid-on.
Bangladesh struck again with the dangerous Nat Sciver-Brunt (two) being trapped leg-before by Fahima Khatun in the eighth over.
England suffered another blow when Knight (six) tried to go for a drive down the ground but missed it completely to be bowled out by Ritu Moni.
