The security men, some on duty at the main Sher-e-Bangla stadium and others at the teams' hotel in downtown Dhaka, fell ill after eating chicken biryani supplied by a local contractor.
"Around 80 percent of them were admitted to the hospital and are getting saline drips," said Mirpur Police Chief Qazi Wajed Ali.
"The contractor who supplied the food has been arrested," he added.
Ali added that reinforcements have been put into place and there was no threat to the players' security.
Bangladesh had ramped up security to 'state-level' for the two quarter-finals to be held in Dhaka on Wednesday and Friday.
The West Indies take on Pakistan on Wednesday with Bangladeshi officials keen to avoid a repeat of the embarrassing March 4 incident when an angry mob pelted the West Indies team bus with stones.
The second quarter-final on Friday features South Africa against New Zealand.
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