The English were also thoroughly beaten by Australia and New Zealand in their first two games.
New Zealand, Australia and Sri Lanka look likely to emerge from Pool A but the final spot in the knockout stages is up for grabs.
Bangladesh haven't looked like world-beaters at this World Cup but providing they don't slip up against any minnows, they could steal a spot in the quarter-finals with a win over England.
England's batting line-up looks thin with the struggling Gary Ballance at No 3, while skipper Eoin Morgan also looks a man who is low on confidence.
Their total of 309-6 today came courtesy of Joe Root who carved a career-best 121 when his team needed him to deliver.
It was a well-constructed knock, full of patience, and he hit the go button when he needed to. Opener Ian Bell added 49.
Such a score should have been enough at Wellington Regional Stadium but England's bowlers never created enough pressure, while Morgan appeared low on ideas with his captaincy.
When his men were struggling to capture wickets, the skipper preferred to sit back and trot through the overs with his spinners who couldn't slow the run rate down.
Thirimanne, 25, suffered some early jitters and had a lucky escape when he was dropped on 98 but played a reliable hand as he registered his fourth one-day ton and the highest score of his career.
Sangakkara, in his 401st one-day international, proved once again why he is one of the best players to have pulled on a pair of pads.
The 37-year-old's stroke play seemed effortless and even James Anderson, the world's fourth-ranked ODI bowler, was shown little mercy. He brought his century up in 69 balls and finished the job in style.
Scoreboard
Manhattan/Worm
Wagonwheel
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