Laxman was on 60 when he developed back spasms and called for a runner after being treated by team physiotherapist Nitin Patel on the field.
The stylish Hyderabad batsman, later named man of the match, followed his 56 in the first innings with his 16th Test century.
Laxman was also involved in an important fifth-wicket partnership of 109 with Sachin Tendulkar, who scored 54 before being dismissed by Suraj Randiv as the off-spinner captured his first five-wicket haul in Test cricket.
Randiv, who came in to the Sri Lanka side for retired Muttiah Muralitharan, took every Indian wicket to fall in the second innings.
Sri Lanka missed a golden opportunity to remove Tendulkar cheaply in the morning session of the final day when he was put down by Tillakaratne Dilshan at forward short-leg with his score on 18.
The prolific batsman and cricket's most-capped Test player made Sri Lanka pay for that early lapse by recording his 56th Test half-century.
Sri Lanka's only success of the morning was the wicket of nightwatchman Ishant Sharma, who was caught at short mid-wicketfor four off Randiv after India had resumed on 53-3.
Sri Lanka won the first Test by 10 wickets in Galle and the second Test in Colombo was drawn.
Mahendra Singh Dhoni, the Indian captain, said Saturday's win answered critics who wrote off the team after the first Test.
"There was a lot said about us when we lost the first Test, that our bowling was weak and the batsmen were not doing enough," he said.
"But we proved in this game that there was nothing wrong. Laxman and Tendulkar were outstanding. Their experience helped us to soak in the pressure of a fourth innings chase ... In the end, it was a very convincing win."
Kumar Sangakkara, his Sri Lankan counterpart, conceded that Laxman took the match away from the hosts, but rued the lack of support for Randiv.
"Laxman was fantastic and Sachin also played really well, but it would have helped if someone had backed Randiv," he said.
"The guys can be proud of their effort all through the series. In this game we were found wanting with our batting, especially in the second innings.
"But I am happy the way we played in the three Tests. It was a young bowling attack and this series was a good learning experience for them."