Ireland reached 286-8 at stumps on the third day to lead the hosts by 131 runs after resuming on a precarious 27-4 in the one-off Test.
Tucker made 108 off 162 balls before falling to the bowling of Ebadot Hossain, with Shoriful Islam taking the catch at extra cover.
The wicketkeeper-batsman hit 14 fours and a six and was brilliantly assisted by Harry Tector and McBrine in partnerships of 72 and 111 to put the pressure back on Bangladesh.
"It is obviously pretty special. It wasn't something that I thought would happen today," Tucker said.
"We were under a lot of pressure. We tried to take it ball-by-ball and bat for as long as we could. It was very special to get a personal reward.
"The team is also in a great position. We are looking forward to tomorrow," he said.
Tector resumed from overnight for a 28-run partnership with Peter Moor before left-arm pacer Shoriful gave the hosts their first breakthrough of the day.
Moor was gone for 16 but Tector and Tucker staved off further damage with their stand.
Tector departed after he was found leg before wicket off Taijul Islam, but his 56 made him the first Irish cricketer to hit half-centuries in both innings of a Test match.
Tucker brought up his century with a cover drive for four off left-arm spinner Taijul, raising his bat after hugging McBrine.
Mark Adair was next to bat, accompanying McBrine in a 31-run stand before he was removed by Taijul, the most successful Bangladesh bowler of the innings with 4-86.
"No one said Test cricket's gonna be easy," said Bangladesh bowling coach Allan Donald.
"I thought they grinded out the day beautifully and yes, once again, one massive tick to Ireland. They fought hard today."
Kevin O'Brien is the only other Irishman to hit a Test century, scoring 118 against Pakistan during Ireland's 2018 maiden Test in Dublin.
Ireland have lost all three of their previous Tests since becoming a full member of the International Cricket Council in 2017.
