Tamim contributed 103 from 100 balls to a Bangladesh record first wicket partnership of 185 with Imrul Kayes (75) after his team had been asked to follow on.
Junaid Siddique and Jahurul Islam (46) consolidated with a composed third-wicket partnership of exactly 100 which enabled the visitors to reach 328 for five at the close, a lead of 105 with a day's play remaining.
England should still wrap up the match comfortably on Monday but Bangladesh have proved thoroughly worthy opponents five years after losing their only other Test at Lord's by an innings and 261 runs within three days.
Tamim reached his century from 94 balls, the fastest in Tests by any Bangladesh batsman and the quickest at Lord's since Indian Mohammad Azharuddin's 87-ball hundred in 1990.
He was particularly severe on off-spinner Graeme Swann, England's best bowler over the past two years, slog-sweeping two sixes and cover-driving a four from one over.
Inevitably with such an attacking approach, Tamim rode his luck, missing completely with several full-blooded drives.
SCORE BOARD
ENGLAND (I Innings): 505
BANGLADESH (I Innings, O/n: 237/7)
Tamim (run out) 55
Kayes c Strauss b Finn 43
Siddique c Prior b Finn 58
Jahurul c Prior b Anderson 20
Ashraful lbw Finn 4
Shakib c Strauss b Anderson 25
Rahim b Finn 16
Mahmudullah b Anderson 17
Shahadat b Anderson 20
Rubel c Cook b Bresnan 9
Robiul (not out) 9
Extras (LB-2, W-3, NB-1) 6
Total (all out, 93 overs) 282
Fall of wickets: 1-88, 2-134, 3-179, 4-185, 5-191, 6-221, 7-234, 8-255, 9-266.
Bowling: Anderson 31-6-78-4, Bresnan 24-5-76-1, Finn 25-5-100-4, Swann 11-6-19-0, Trott 2-0-7-0.
BANGLADESH (II Innings)
Tamim c Trott b Finn 103
Kayes c Bell b Finn 75
Siddique (batting) 66
Jahurul c&b Trott 46
Ashraful c Prior b Anderson 21
Shahadat b Bresnan 0
Shakib (batting) 2
Extras (B-4, lb-8, W-2, nb-1) 15
Total (for 5 wkts, 85 overs) 328
Fall of wickets: 1-185, 2-189, 3-289, 4-321, 5-322.
Bowling: James Anderson 20-6-66-1, Tim Bresnan 20-5-88-1, Steven Finn 16-4-66-2, Graeme Swann 25-4-80-0, Jonathan Trott 4-0-16-1.
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