After polishing off the Indian tail early in the morning, Australia rode on David Warner's pacy 40 and Rogers' patient 69 to extend their lead before off-spinner R Ashwin picked up two wickets at the MCG here.
Rain intervened after lunch, resulting in a loss of 85 minutes of play and when play resumed, regular wickets fell to leave the hosts at 174/4 in 43 overs at tea, which was taken after an extended second session.
At the Melbourne Cricket Ground, in the morning session India were bowled out for 465 runs in their first innings, handing the hosts a 65-runs lead.
At tea, Shaun Marsh (15*) and debutant Joe Burns (8*) were the unbeaten batsmen.
After play resumed at the MCG, Rogers (69 runs, 123 balls, 8 fours) and Shane Watson (17) tried to increase their lead further and they were nearly helped by Shikhar Dhawan, who dropped a tough one-handed chance at second slip, diving in front of R Ashwin (2-21) at first slip.
Steve Smith (14) came out to bat and took Australia past the 100-run mark in the 26th over adding 33 runs for the third wicket. Meanwhile, Rogers went on to score his fourth successive half-century of the series, getting to the mark off 81 balls, in the 28th over.
Four overs later his partnership with Smith was broken by Umesh Yadav (1-55) when the young skipper was caught by Ajinkya Rahane as the leg slip finally came into play.
Marsh and Burns then negotiated the rest of the session without damage.
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