It is only the fourth time in history England have won there, and it is the first time since David Gower's tourists were triumphant in 1984-85.
Jonathan AgnewBBC cricket correspondent
"There were handshakes, hugs, embraces and slaps on the back all round in the England dressing room. They have worked so hard in this series. What odds were there on us witnessing that? It's quite remarkable. They have done it the hard way, through outstanding leadership on and off the pitch."
England also inflicted India's first series loss on home soil since the legendary Australia team of Shane Warne, Glenn McGrath and Ricky Ponting won in 2004 - a series which became billed as
"The Last Step To Greatness".
Cook's team may not be "great" yet, but they certainly gave players like Sachin Tendulkar, Virender Sehwag and Mahendra Dhoni plenty to think about with their dominant performances.
Trott and Bell were the two players most in need of a good score on the final day of the match - and both delivered emphatically.
England players on Twitter
"A truly amazing all round effort to win out here. Everyone has been brilliant."
"That is a monumental win for us. First time in India during my lifetime. Proud to play a small part in it."
"First Test tour, Test debut, first series win - doesn't get any better than this!"
"A day and tour I shall never forget! Absolutely brilliant!"
"Where's that tweet predicting 4-0 win to India Michael Vaughan? The boys can't seem to find it? #turnsroundquickly"
The Warwickshire pair posted their highest Test scores of the year, with a century particularly welcome for Bell who averaged just 18 in India before he came to the crease when England were stuttering slightly at
94-3 on the fourth day.
Scoring may have been slow, with Trott facing 310 balls and Bell 306, but it is a measure of England's progress that they suffered so few alarms on the final day after such a chastening start to the year.
But there was no way Trott was going to allow the year to end on a sour note. He continued in a defiant manner on the fifth morning - timing the ball to perfection - as he eased a four through the leg side to bring up his eighth Test century.
It appeared he might bat throughout the final day until his dismissal just before tea. The 31-year-old chased a Ravichandran Ashwin delivery and chipped the ball into the hands of Virat Kohli at leg-slip.
Bell batted similarly serenely, but did enjoy a let-off on 75 when he slashed hard outside off stump and was dropped at slip by Sehwag.
He capitalised on his fortune late in the day when he completed his 17th Test century - the slowest of his international career - as he and Root, who swept Ashwin for six to compound the off-spinner's misery, closed out the historic and memorable series victory.
England, who consolidated their second-placed position behind South Africa in the ICC Test rankings, travel to New Zealand for a three-Test series in March before hosting the Kiwis in the summer ahead of back-to-back Ashes contests.
We are using archive pictures for this Test because several photo agencies, including Getty Images, have been barred from the ground following a dispute with the Board of Control for Cricket in India, while other agencies have withdrawn their photographers in protest.
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