Smith turned in figures of 2-11 from her four overs and played the significant role in captain Nigar Sultana's run-out as Heather Knight's side limited the Bangladeshis to 7-97 in Saturday's group B showdown under the lights in Sharjah.
Smith and Charlie Dean each claimed two wickets to help successfully defend a modest target after Danni Wyatt-Hodge had scored 41 in their 7-118 on a tricky pitch.
"It was what we prepared for but we've just got to find a way to do it sometimes," shrugged England captain Heather Knight after the hard-earned victory.
"It's not going to be beautiful and it's not going to be sexy all the time and if we do get these conditions again, we're going to have to graft a little bit and try to minimise the dot-balls because boundaries are going to be really hard to hit."
England won the toss and opted to bat, with opener Maia Bouchier enjoying a reprieve when she was dropped by Rabeya Khan at point off the bowling of Marufa Akter on 16.
Rabeya redeemed herself when she had Bouchier caught by Nahida Akter at mid-on for an 18-ball 23 in the seventh over with 48 on the board.
Nat Sciver-Brunt soon followed lbw to Fahima Khatun for two and, with the Bangladesh bowlers putting the brakes on England after a return of 0-47 in the power play, Knight was bowled by Ritu Moni for six to leave her side on 3-73 after 11.2 overs.
Wyatt-Hodge quickly followed, stumped by Sultana off Nahida after hitting five boundaries in her 41, and Alice Capsey was caught for nine reverse-sweeping.
Dani Gibson and Charlie Dean both fell cheaply as England were restricted to 118 despite Sophie Ecclestone's two-ball, eight-run cameo with Fahima's 2-18 the pick of the bowling contributions.
Dean trapped Dilara Akter in front for six in her first over before fellow opener Shathi Rani chipped Smith to Ecclestone in the fifth over, leaving struggling Bangladesh to rack up just 20 powerplay runs.
Skipper Sultana's more adventurous approach guided her side to 2-42 by the halfway point, but when she was run out for 15 from Smith's accurate throw to wicketkeeper Jones, the game was up.
Legspinner Sarah Glenn clean bowled Shorna Akter for two in the 13th over, and although Sobhana Mostary pulled Dean for six in the 16th, Smith mopped up Taj Nehar for seven and Sciver-Brunt accounted for Ritu for two to leave them needing 34 from 14 balls.
Mostary's 44 from 48 balls before she fell lbw to Dean was the only score of note in an innings which was never really allowed to get going.
England are likely to face a tougher test when they meet South Africa in Sharjah on Monday, when Knight knows they will have to fight fire with fire.
"They had a convincing win the other night, didn't they?" she said.
"They batted really well. They're always competitive, they always bring that fire and passion, so we're going to have to play really well."
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