Finn, 21, who made his debut on the two-Test tour of Bangladesh in March, finished with five for 87 from 24 overs of accurate, probing pace bowling from the Pavilion End. He took four for 100 in the first innings.
Set 160 to win in two full sessions, England reached their target in the last over before tea. Captain Andrew Strauss began at breakneck speed with four boundaries off the first two overs and reached his half-century from as many balls.
After making 83 in the first innings on his return to the England side he was out for 82 from 88 balls, edging a catch behind off his opposite number Shakib Al Hasan.
Bangladesh lost their remaining five wickets in the morning for only 54 runs, a disappointing performance following their exuberant batting on Sunday which looked likely to salvage a draw after they had lost each of their previous six Tests against England.
Finn struck three times in 15 balls after play resumed on a chilly, overcast morning with Bangladesh on 328 for five, removing Shakhib (16), Junaid Siddique (74) and Mushfiqur Rahim (0) in consecutive overs.
SCORE BOARD
ENGLAND (I Innings): 505
BANGLADESH (I Innings: 282 & O/n: 328/5)
Tamim c Trott b Finn 103
Kayes c Bell b Finn 75
Siddique c Bresnan b Finn 74
J Islam c&b Trott 46
Ashraful c Prior b Anderson 21
S Hossain b Bresnan 0
Shakib c Morgan b Finn 16
Rahim c Prior b Finn 0
Mahmudullah c Prior b Bresnan 19
R Hossain c Strauss b Bresnan 4
R Islam (not out) 0
Extras (B-7, LB-14, W-2, NB-1) 24
Total (all out, 110.2 overs) 382
Fall of wickets: 1-185, 2-189, 3-289, 4-321, 5-322, 6-347, 7-354, 8-361, 9-381.
Bowling: Anderson 29-8-84-1, Bresnan 26.2-9-93-3, Finn 24-6-87-5, Swann 27-5-81-0, Trott 4-0-16-1.
ENGLAND (II Innings)
Strauss c Rahim b Shakib 82
Cook lbw Mahmudullah 23
Trott (not out) 36
Pietersen (not out) 10
Extras (LB-5, W-1, NB-6) 12
Total (for 2 wkts, 35.1 overs) 163
Fall of wickets: 1-67, 2-147.
Bowling: S Hossain 2-0-19-0, R Islam 1-0-12-0, Shakib 16-1-48-1, R Hossain 1-0-8-0, Mahmudullah 15.1-1-71-1.
