Chasing a target of 179, England had collapsed to 82-6 but former captain Knight struck an unbeaten 79 as her team eventually reached the target in the 47th over.
Sophie Ecclestone had earlier taken 3-24 as Bangladesh were restricted to 178 all out with Sobhana Mostary (60) and Rabeya Khan (43 not out) ensuring Asian side breached the 170-run mark.
Having won the toss and electing to bowl for the second successive World Cup match, Nat Sciver-Brunt's side made a fast start, reducing Bangladesh were reduced to 25-2 inside Six overs.
Lauren Bell (1-28) and Linsey Smith (2-33), accounted for Rubya Haider for four and skipper Nigar Sultana Joty for a second-ball duck respectively.
England's economical bowling succeeded in stifling the Tigresses' innings and two brilliant catches from Amy Jones standing up to the stumps left Bangladesh 85-4 in the 23rd over.
From there, England's spinners turned the screw on the opposition as Ritu Moni holed out to deep midwicket for five to give Dean her second dismissal, having previously accounted for Shorna Akter.
Whilst wickets tumbled around her, Sobhana looked calm and composed as her maiden international fifty helped her side fight back against England's spin barrage.
Ecclestone took two wickets in consecutive overs to take her wicket tally to three having removed Sharmin Akhter for 30, as Bangladesh were reduced to 130-7 with just seven overs remaining.
The 26-year-old slow left-armer knocked over Fahima with a flighted delivery for seven before tempting Nahida Akter into an ill-fated drive which found Dean close in.
Rabeya offered valuable late-order resistance for Bangladesh, adding an unbeaten 43 from 27 balls which featured six fours and one towering six off Smith.
But once Sobhana was given out lbw for a fine 108-ball 60 off the bowling of Alice Capsey (2-31), Bangladesh were eventually bowled out for 178.
England's reply hit turbulence early on as Marufa Akter pinned Jones leg before wicket in the first over and then Tammy Beaumont in the seventh to leave the score 29-2.
The arrival of Sciver-Brunt looked to have steadied the ship, as, alongside Knight, who had been given reprieves from being given out caught and lbw, the pair put on 40 for the third wicket.
But Fahima Khatun spun the game back in Bangladesh's favour with three quick wickets, including Sciver-Brunt for a well-made 32, to leave England 78-5.
Capsey looked assured as she and Knight took England past 100, but the 21-year-old was prized out for 20 by Shanjida Akther Maghla soon after.
Knight's majestic, lofted drive over Maghla's head showed she was up for the fight and when she reached her half century, England needed just 41 more to win.
Charlie Dean provided able support to Knight as the duo's partnership was the highest of the match, and it was the former who sealed the game with a glorious cover drive to the boundary.
The Proteas Women made it five consecutive victories at the ICC Women's World Cup after a comprehensive 150-run win (DLS method) against Pakistan in match that was shortened due to rain.
Sri Lanka successfully defended a measly 202 to win their ICC Women's World Cup encounter against Bangladesh by seven runs in Mumbai on Monday night.
Phil Salt and captain Harry Brook struck half-centuries while Adil Rashid took four wickets to help England beat New Zealand by 65 runs in the second T20I in Christchurch.
The Proteas Women have secured a spot in the ICC Women's World Cup semifinals after the game between New Zealand and Pakistan was abandoned because of rain in Colombo.
