Najmul hit 117 from just 93 in Chelmsford in front of a crowd of 4,000 made up largely of Tigers fans.
The win gave Bangladesh a 1-0 lead in a three-game run after a rain-stained lack of result on Tuesday at the English county seat of Essex.
Bangladesh veteran Mushfiqur, with four runs required on four balls, reached a limit and completed 36 steps.
Najmul, who eventually beat Curtis Campher, shared a fourth wicket of 131 with Towhid Hridoy (68).
In a match rain reduced to 45 overs per team, Tector’s fourth century in his last nine ODI innings took Ireland to 319-6.
The 23-year-old hit seven fours and 10 sixes from just 113 balls, with George Dockrell hitting 74.
Bangladeshi pacemaker Shoriful Islam conceded 83 runs in nine overs in an innings where Ireland broke a national record 16 sixes.
This series ends on Sunday.
Tuesday’s elimination, something Ireland had hoped to avoid by moving the series to England, robbed the Irish of the 3-0 series they needed to secure an automatic entry into this year’s World Cup in India.
Ireland will now have to take part in the June 18-July 9 qualifying tournament in Zimbabwe, which also includes former champions West Indies and Sri Lanka.