Najmul made 117 off just 93 balls at Chelmsford in front of a 4,000-strong crowd largely made up of Tigers fans.
Victory gave Bangladesh a 1-0 lead in a three-match series after Tuesday’s rain-marred no-result at English county Essex’s headquarters.
Bangladesh veteran Mushfiqur, with four runs required off four balls, hit a boundary and finished 36 not out.
Najmul, who eventually holed out off Curtis Campher, shared a fourth-wicket stand of 131 with Towhid Hridoy (68).
In a match reduced by rain to 45 overs per side, Tector’s fourth century in his last nine ODI innings took Ireland to 319-6.
The 23-year-old struck seven fours and 10 sixes in a mere 113 balls, with George Dockrell making 74.
Bangladesh paceman Shoriful Islam conceded 83 runs in nine overs during an innings where Ireland struck a national record 16 sixes.
This series concludes on Sunday.
Tuesday’s washout, something Ireland hoped to avoid by moving the series to England, denied the Irish the 3-0 series clean sweep they needed to secure automatic entry into this year’s World Cup in India.
Ireland will now have to compete in the June 18-July 9 qualifying tournament in Zimbabwe, that also includes former champions West Indies and Sri Lanka.
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