Warner smashed 101 off just 114 balls while Rogers fell five short of a hundred as the Australian pair put on an even 200 for the opening wicket at the Sydney Cricket Ground.
Both were dismissed after lunch but new captain Steve Smith (81 not out) and Shane Watson (56 not out) ensured the hosts maintained their momentum with an unbroken third-wicket stand of 144 to take Australia to 348 for two at stumps.
Although Australia have already secured the Border-Gavaskar Trophy after wins in Adelaide and Brisbane, emotions were heightened for the final match because it is being played at the same ground where former team mate Phillip Hughes was fatally injured six weeks ago.
The death of the 25-year-old Hughes, who was struck by a short delivery in a first class match in late November, has hung over the entire series but has been brought to the fore in the New South Wales state capital.
Cricket Australia unveiled a memorial plaque outside the home side's dressing room on Monday, while both teams paid tribute to the left-hander before the game began on Tuesday.
Regular captain Michael Clarke, who tore a hamstring in the first test and is fighting to be fit for the upcoming World Cup, paid his own tribute to Hughes before the match.
"This is where he played his last game, this is where his spirit will live forever," Clarke told the Nine Network.
Rogers and Warner did exactly that after Smith, Clarke's replacement as skipper, won the toss and chose to bat on a flat pitch offering little assistance to the Indian bowlers.
BLISTERING START
They put on 123 in a blistering opening session with Warner pausing to kiss the pitch and signal to the sky when he reached 63, the score Hughes was on when he was hit by the ball.
Warner, who scored twin centuries in Adelaide, brought up his ton when he pulled Mohammed Shami to the deep square leg for his 16th boundary.
But he departed shortly after when he edged Ravichandran Ashwin to Murali Vijay at gully after he and Rogers had brought up their double-century partnership.
Rogers, who had made half-centuries in each of his last four test innings, was given a life on 19 when he was dropped by Lokesh Rahuon at second slip but once again missed out on triple figures when he got a thick edge from Mohammed Shami that crashed into his stumps.
Smith has already scored three tons in the series and continued his rich form by notching up his 10th half-century in tests as the hapless Indian bowlers went wicketless in the final session.
Watson, who has failed to convert his starts in the series, looked solid on his way to his 24th test fifty but was lucky to survive an edge in the last over off Umesh Yadav which flew through the hands of Ashwin at first slip.
