The wicketkeeper-batsman made 108 off 162 balls before falling to the bowling of Ebadot Hossain, with Shoriful Islam taking the catch at extra cover.
The 26-year-old Tucker hit 14 fours and a six during his time at the crease.
He was brilliantly assisted by Herry Tector and Andy McBrine in partnerships of 72 and 111 after the tourists resumed on a precarious 27-4 in their second innings in the one-off Test.
Tector's 56 made him the first Irish cricketer to hit fifties in both innings of a Test match.
Tucker brought up his century with a cover drive for four off left-arm spinner Taijul Islam, raising his bat after hugging McBrine.
Kevin O'Brien is the only other Irishman to hit a Test century.
He scored 118 against Pakistan during Ireland's 2018 maiden Test in Dublin.
