After setting a daunting victory target of 386, India managed to dismiss the Sri Lankans for 268 on a wicket which slowed down considerably to wrap up the series 2-1, their first overseas series victory since June, 2011 against the West Indies.
It was new captain Virat Kohli's first series win ever since he took over reins of the team from Mahendra Singh Dhoni.
On an absorbing final day's play, the visitors took two quick wickets in the morning session before Mathews (110) and debutant Kusal Perera (70) scripted a spirited fightback to frustrate the Indians.
The duo put on a resolute 135-run partnership for the sixth wicket, the second-highest stand in the fourth innings in Sri Lanka's Test history. Roy Dias and Duleep Mendis hold the record of biggest fourth innings partnership with a 214- run stand.
Off-spinner R Ashwin was the pick of the Indian bowlers with four for 69 while paceman Ishant Sharma scalped three for 32. Umesh Yadav chipped in with two for 65.
The last time an Indian team won a series in Sri Lanka was way back in 1993 under Mohammed Azharuddin's leadership when they sealed the series with 1-0 scoreline.
The fifth day's action was not devoid of drama as more on-field altercation took place with Kusal Perera being involved in a heated exchange with Indian players.
Ashwin took the last wicket of Nuwan Pradep to bring about India's moment of glory, triggering scenes of wild celebrations in the dressing room as the players hugged each other and picked up the stumps as mementos.
Ashwin provided the vital breakthrough by breaking the sixth-wicket partnership with Perera's scalp. The second new ball, taken after tea break, turned the tide completely in India's favour with Ishant getting rid of Mathews in the very first over.
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