Tamim, who made 128, took advantage of a depleted West Indies attack, operating on an unresponsive pitch in sweltering temperatures, to smash 17 fours from 243 balls in close to five and a half hours of batting to give Bangladesh the platform for a lead of 252 at the stumps on the fourth day on Sunday.
Another middle-order collapse however, undermined Bangladesh's progress in the final session before Shakib Al Hasan (26 not out) and Mushfiqur Rahim (28 not out) batted through the last 90 minutes to share an unbroken stand of 54 for the sixth wicket to get them back on track.
Tamim held centre stage and arrived at his milestone from 206 balls, when he guided a delivery from off-spinner Ryan Austin into square cover, and scrambled a single as a bout of cramp surged through his right-leg.
Fortune had favoured Tamim twice in his innings though. He was 30 when Floyd Reifer dropped him at first slip off Darren Sammy, and he was 76, when Omar Phillips muffed a straightforward chance at mid-wicket off Austin.
The left-handed opener, however, kept his nerve, and anchored two vital stands. He shared an opening stand of 82 with Imrul Kayes, who scored 24, and then added 146 for the second wicket with Junaid Siddique, who made 78.
'It's a dream come true for any player to score a hundred in Test match cricket,' said Tamim.
'Our head coach Jamie Siddons always says that 'the more balls we can leave, the more our chances increase of us getting a big score.' In other words, it is not always about the shots. The good balls you respect, and the bad balls you put away.
'The pitch is very flat, and if you do not want to get out, no one can get you out. It is a really flat pitch. I was going well, but then I got out playing a poor stroke.'
He added: 'We are going very well right now, but we lost four wickets in the last session which set us back a bit. Mushfiqur and Shakib are doing a good job though, and if we can get a lead of around 300, I think it would be enough to put them under pressure.
'Shakib is also one of the best spinners in the world, and I think he and the other bowlers can make it hard for the West Indies on the last day.'
Shakib and Rahim eased the late pressure on Bangladesh, after the Tigers lost Siddique, Mohammad Ashraful, and Raqibul Hasan for nine runs in the space of 16 balls to slide to 267 for five.
Siddique batted solidly throughout his four-hour stay at the crease, and reached his 50 from 119 balls, when he cut Austin to deep backward point for a deuce.
But in the hunt for quick runs in the final session, he was caught at gully off Sammy before Mohammad Ashraful was adjudged lbw to Kemar Roach for three, and Raqibul was bowled by Sammy for 18.
This was in stark contrast to what transpired before lunch, when Bangladesh made steady progress to reach 109 for one, after they had continued from their overnight total of 26 without loss.
Tamim and Kayes formally took Bangladesh to the lead - after they trailed West Indies by 69 on first innings - when Austin delivered a no-ball in his sixth over.
Tamim then reached his 50 from 72 balls, when he turned a delivery from Sammy into mid-wicket for a single. But soon after Kayes was caught at extra cover driving at a flighted ball from Austin about an hour before lunch.
After the interval, Tamim and Siddique batted through the entire afternoon session to carry Bangladesh to 200 for one at tea with both reaching their milestones.
After the break, Bangladesh looked to increase the tempo of their scoring, but they lost Tamim caught at mid-on about 20 minutes after the break.