Chasing an improbable 415-run victory target, the hosts resumed on 67/2 and rode on Rahim's gutsy 101 and Tamim Iqbal's gritty 52 before folding for 301 in 75.2 overs.
Amit Mishra (4/92), Ishant Sharma (3/48), and Zaheer Khan (2/90) struck at regular intervals to bowl out the hosts at the Zohur Ahmed Chowdhury stadium.
The baby-faced Rahim threw his bat at everything hurled at him but that was merely a desperate attempt by a number seven batsman to delay the inevitable.
The diminutive stumper-batsman took 112 balls to hit his maiden Test ton -- also the fastest by a Bangladeshi batsman -- and was the last man out.
Both the team now moves to Mirpur for the last Test starting on Sunday.
Rahim, who came in just before lunch when his side were 135/5, took the Indian attack by the scruff of its neck, hitting 17 fours and a six.
With this win, India maintained their unbeaten record in Bangladesh. They have now won five out of six matches in Bangladesh, including a drawn Test in 2007.
The Test, which began with a war of words between the two sides after India's stand-in captain Virender Sehwag had described the hosts as an ordinary side, was also a memorable one for opening batsman Gautam Gambhir who became the only fourth batsman to have scored five successive Test tons.
Sachin Tendulkar also reached the milestone of becoming the only player to score 13,000 Test runs by notching up his 44th hundred, which also earned him Man of the Match award.
An Indian batting collapse on the opening day of the weather-hit match provided some excitement but the Test ultimately ended on expected lines with the visitors scripting a comfortable win.
Chasing a stiff 415, Bangladesh needed 348 from the final day and once they were 136/5 at lunch any hopes of an improbable win had evaporated.
Wickets fell at regular intervals with the Indian bowlers applying pressure all day with some impressive show till came the tail.
Using the early movement of the ball due to foggy conditions, Ishant struck twice in the first hour conceding just 11 runs from his seven-over uninterrupted spell to give his side the stranglehold in the Test.
Ishant first removed key batsman Mohammad Ashraful (27) in the fourth over of the day with Rahul Dravid taking a fine catch at the second slip.
Bangladesh were 94/4 soon with Ishant striking again in his sixth over, trapping Raqibul Hossain LBW.
Sehwag introduced himself in place of S Sreesanth and immediately removed Tamim Iqbal to reduce Bangladesh to 135/5.
Overnight batsman Tamim went for an unnecessary slash off a Sehwag delivery which shaped away after pitching on the off and the thick edge went straight to Dravid's safe hands at slip. Tamim, who resumed the day on 23, faced 122 balls and hit six boundaries in his 52.
Bangladesh, who resumed the post-lunch session on 136/5, showed no intention of digging their heel and trying to save the match.
Captain Shakib Al Hasan perished in the third over after lunch, foxed by an Amit Mishra googly which took the shoulder of his blade for Sehwag to take a head-high catch at silly point.
Mahmadullah (20) hit two fours in the same over of Zaheer Khan but the veteran Indian bowler had the last laugh as he removed the batsman in his next over with wicketkeeper Dinesh Karthik collecting the faint edge.
Mishra struck twice in quick time to remove Shahadat Hossain (24) and Shafiul Islam (8) before he went on to end Rahim's heroics.
SCORE CARD
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Fall Of Wickets 1/8 (Imrul Kayes, 2.3 ov.), 2/47 (Shahriar Nafees, 13.5 ov.), 3/79 (Mohammad Ashraful, 21.2 ov.), 4/97 (Raqibul Hasan, 29.3 ov.), 5/135 (Tamim Iqbal, 40.5 ov.), 6/145 (Shakib Al Hasan, 45.4 ov.), 7/170 (Mahmudullah, 52.5 ov.), 8/230 (Shahadat Hossain, 63.5 ov.), 9/258 (Shafiul Islam, 67.6 ov.), 10/301 (Mushfiqur Rahim, 75.2 ov.) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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