The Tigers made a decent start with Tamim Iqbal and Soumya Sarkar making 56 for no loss but the latter then fell to Ben Stokes, lifting one high onto the offside for substitute fielder Jonny Bairstow to take.
Mark Wood's catch to get rid of Imrul Kayes.
The stylish Tamim hit 12 boundaries and three sixes in his innings while Rahim struck eight boundaries in his 72-ball knock.
Sabbir Rahman scored a fluid 24 off 15 as he took to Ball, before swiping a shot straight to Jason Roy off Plunkett in the final over.
He and Hales milked the Bangladesh bowlers for all they were worth, barely breaking sweat in the process, a barrage of ones and twos punctuated by the odd drive, cut or pull to the fence.
Root and Morgan shared 143 for the third wicket, while the Yorkshireman showed deftness and intelligence to post his highest ever score in a one-day worldwide.
Woakes, who came into the match following a thigh problem and could now be replaced by next-in-line Steven Finn if this side strain proves serious, was not the only injury worry for England, with Root hobbling during his 10th one-day worldwide century. Skipper Mashrafe Mortaza said,"We could not get wickets in the middle overs". From then on it was the Joe Root and Eoin Morgan show as they showed patience as well as aggression to take them past Bangladesh's total with 3 overs to spare. A side strain is a tricky injury for a seam bowler to shake off and can easily be made worse by trying to play through the pain.
England were rarely troubled during their run-chase and Root, England's recently-appointed Test captain, said: "We are a (one-day) side that are getting better at it (chasing totals in excess of 300). [Tamim catch?] I think the umpires' decision is final". The Warwickshire man is useful with both bat and ball, so when he had to throw in the towel after only two overs, England's attack suffered.
Morgan, who was able to watch from the other end as Root completed his 10th ODI hundred, said: "Obviously Alex continued his form - Joe, unbelievable, as usual - and it was nice for me to get some runs as well".
"I haven't seen a team win a global tournament playing defensively", said the Australian.
"I'm not saying we're outright favourites, there are a lot of good teams that can beat anyone on their day".
"It is a great statement of intent and it fills us with confidence going into the next game".
The development of Ben Stokes into a top-class allrounder has been key, but this is a side well-served by all-rounders - Chris Woakes, Moeen Ali, Adil Rashid and Liam Plunkett could also fit in this category - meaning they bat deep and have plentiful bowling options. Bangladesh Cricket team also going to try their best against the big boys to give a surprise for their country and team fans. "We'll assess everything in the morning".
Roy, so unequivocally backed by Morgan to open for the entire tournament despite his lack of runs, had to reflect on his fifth single-figure score in his last six ODI innings after paddling an attempted ramp-sweep just within the clutches of short fine-leg - where Mustafizur Rahman held a memorable leaping catch above his head.
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