Nayeem, who was 17 years 355 days old at the start of the game, finished with 5-61, as Bangladesh bowled out the West Indies for 246 to take a 78-run first innings lead.
But the hosts were unable to cash in on the advantage, slumping to 55 for five in their second innings at stumps on the second day for a lead of just 133 runs.
Jomel Warrican and Roston Chase each took two wickets to help the West Indies claw their way back into the contest.
Mushfiqur Rahim remained unbeaten on 11 alongside Mehidy Hasan, on what looked like a batting minefield at the Zahur Ahmed Chowdhury Stadium.
Nayeem, who surpassed Australian Pat Cummins's five-wicket haul on debut at 18 years and 193 days, took care of the West Indies lower middle-order after senior spinners provided the initial breakthroughs.
Skipper Shakib Al Hasan, who finished with 3-43, dismissed Shai Hope and Kraigg Brathwaite at the start of his spell after Taijul Islam removed Kieran Powell for 14 to help the hosts reduce West Indies to 31-3.
Roston Chase and Sunil Ambris tried to rebuild the innings but Nayeem soon struck, removing both batsmen in successive overs for 31 and 19 runs respectively.
Shimron Hetmyer then launched a counter-attack to score 63 off 47 balls with five fours and four sixes for a brief respite from the spin onslaught.
Hetmyer shared 92 runs with Shane Dowrich to lift West Indies from a precarious 88-5 but the side collapsed again once Mehidy broke the stand.
Nayeem then dismissed Devendra Bishoo, Kemar Roach and Warrican to complete his five-for before Sakib removed Shannon Gabriel to wrap up the West Indies innings.
Dowrich held one end firm to stay unbeaten on 63, his third Test fifty which came off 101 balls with the help of two fours and three sixes.
Earlier in the day, Bangladesh were bowled out quickly for 324 in their first innings after resuming on their overnight score of 315-8.
Left-arm spinner Warrican took the remaining two Bangladeshi wickets as the hosts added just nine runs.
Bangladesh owed their first-innings score to Mominul Haque, who struck 120 off 167 balls while Imrul Kayes and Shakib made 44 and 34 respectively.
First Published: November 23, 2018, 5:24 PM IST