Sehwag's 293, a robust 100 not out from skipper Mahendra Singh Dhoni and a string of half-centuries by the middle order fired the hosts to their highest ever total of 726-9 declared in reply to Sri Lanka's 393.
The tourists, trailing by 333 runs on the first innings, were 11-0 in their second knock at stumps on the third day on a wearing track at the Brabourne stadium.
Tharanga Paranavitana and Tillakaratne Dilshan survived three overs of spin before close, but Sri Lanka face a daunting task to avoid defeat over the next two days.
India's total surpassed their previous best of 705-7 declared against Australia at the Sydney cricket ground in January, 2004.
Sehwag had moved from his overnight score of 284 to 293 when he tapped a flighted ball from Muttiah Muralitharan back to the bowler who took the catch on second attempt.
A hush descended at the stadium where some 15,000 home fans had packed the stands anticipating a world record by the swashbuckling opener.
Sehwag returned to a warm applause for his 254-ball effort which was studded with 40 boundaries and seven sixes.
Off-spinner Muralitharan's first success after 21 overs costing 124 runs broke a second-wicket stand of 237 between Sehwag and Rahul Dravid that had lifted India to 458-2.
Sehwag's two triple centuries, matching Australian legend Don Bradman and West Indian great Brian Lara, were 309 against Pakistan in Multan in 2004 and 319 versus South Africa in Chennai last year.
Dravid fell soon after for 74, edging a wild drive off seamer Chanka Welegedara to wicket-keeper Prasanna Jayawardene to make it 487-3.
Dravid, who survived a confident appeal for a catch at the wicket off Rangana Herath earlier in the morning, hit five boundaries and a six.
But Sri Lanka's misery under the hot sun did not end as the famed Indian middle order flexed its muscle to lead a run-riot.
Sachin Tendulkar hit 53, sharing a fourth-wicket stand of 71 with Venkatsai Laxman, when he was bowled by seamer Nuwan Kulasekara soon after lunch.
Laxman chipped in with 62 and Yuvraj Singh made 23, before both were dismissed before tea on a dusty pitch that provided turn to the slow bowlers.
Dhoni dominated the last session with a sparkling century that was achieved with a six off Herath over the mid-wicket boundary.
The Indian captain, who was on 50 when the ninth wicket fell, hit 50 off the last 56 runs in the company of number 11 Pragyan Ojha, who made just five.
Dhoni finished with six sixes and three boundaries, closing the innings as soon as he reached his hundred.
All the four specialist Sri Lankan bowlers conceded over 100 runs with Herath being the most expensive with 3-240 from 53.3 overs. Muralitharan finished with 4-195 from 51 overs.
India lead the three-match series 1-0 after an innings and 144-run victory in the second Test in Kanpur last week