Masakadza hit 79 off 53 balls, equalling his own record of a highest individual innings for Zimbabwe, while Sibanda struck 46 off 39 in the comeback.
Bangladesh did well to restrict Zimbabwe in the final few overs after Masakadza and Sibanda broke the team record set by Sikandar Raza and Chamu Chibhabha's partnership of 100 runs against Afghanistan last October.
Zimbabwe looked on course to hit a big innings after skipper Elton Chigumbura won the toss and opted to bat at the Sheikh Abu Naser Stadium in Khulna, known to favour the batting team.
Bangladesh had to wait until the 12th over for their first breakthrough when Sibanda holed out a catch at square leg to Tamim Iqbal off Shakib Al Hasan's delivery.
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Shakib dropped Masakadza on 73 off Mustafizur Rahman, but debutant wicketkeeper Nurul Hasan ensured it did not cost the team much by ending his innings with a brilliant throw.
Rahman then bowled Chigumbura and Luke Jongwe off consecutive balls to finish with 2-18, before Al-Amin Hossain grabbed two wickets in the final over.