The West Indies had been put in to bat and an increasingly regular slow start for Chris Gayle was punished when Mohammad Saifuddin lured the opener into poking at a ball that was moving away from him and a thin edge was snaffled behind the stumps.
Set 322 for victory, Shakib - the No 1 ODI all-rounder in the world, brought up his hundred off only 83 balls and finished unbeaten on 124 having shared an unbroken 189-run third wicket stand with Liton Das (94no), as Bangladesh reached their target with 51 balls spare.
There were huge cheers from the Bangladesh fans around the ground when Hasan became just the second player to score 6,000 career ODI runs for the Tigers.
Bangladesh's highest one-day worldwide run chase put them behind only Ireland's 329-7 against England in 2011 in the history of successful World Cup chases.
Shimron Hetmyer scored a quickfire 50 off 25 balls to hit the joint-fastest fifty of this World Cup.
In a shocking turn of events, Chris Gayle (0) went down on a duck on Mohammad Saifuddin's delivery.
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South Africa were unchanged from the team that started the match against the West Indies, which ended in a no result. Quinton de Kock then fired 68 as South Africa chased down the revised DLS target of 127 in 28.4 overs.
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Earlier, West Indies batsman Shai Hope hit a patient 96 while Evin Lewis and Shimron Hetmyer also made half-centuries to help the former champions post a competitive 321-8 from 50 overs. They are the only Bangladesh batsmen to hit centuries in Cricket World Cups.
And skipper Jason Holder's 33 from 15 balls - including four fours and two sixes - ensured the Windies went past 300.
Bangladesh have had the upper hand against the West Indies in recent meetings, winning six of the last seven, including all three games in the Tri-nation series in Ireland last month.
Taunton is mainly cloudy and dry and the West Indies-Bangladesh match should play out without interruption.