India was all out in its second innings for 112 after chasing what had seemed a modest victory target of 176 in the three-test series opener.
Left-arm spinner Rangana Herath took 7-48 while Ajinkya Rahane top scored for India on 36.
Sri Lanka was bowled out for 183 in its first innings after electing to bat first and India took a lead of 192 runs after piling on 375 in reply.
Sri Lanka was heading toward an innings defeat Friday when it was reduced to 95-5 in its second innings, but an unbeaten 162 by wicketkeeper Dinesh Chandimal revived the hosts' fortunes, and the bowlers then put them on top.
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