However, the left-arm spinner could not prevent the tourists reaching the target with nine balls to spare.
Liton Das set the Bangladeshis on the way to an 11th consecutive ODI win over the West Indies with an even 50 while captain Tamim Iqbal played his part with 34.
They seemed determined to trip themselves up with some careless shots though before an unbeaten 32 from wicketkeeper-batsman Nurul Hasan in partnership with Mehidy Hasan saw them home.
"I was very fortunate to finally get an opportunity on this tour because I was practising and working very hard and I'm glad I made full use of it," said Taijul via an interpreter on receiving the man of the match award.
His victims included top-scorer Nicholas Pooran, the West Indies captain ending a run of low scores in one-dayers with an innings of 73 off 109 balls with two sixes and four fours embellishing his innings.
Only three other players got into double figures in what was yet another sub-par overall batting effort by the Caribbean team.
"I felt I still left some runs out there but I am happy to finally get a decent score in ODIs even though it's obviously disappointing to lose once again," said Pooran.
"We have to lift our heads from here and try to learn from this experience because we have two more tough opponents ahead before we get a break."