Pujara and Saha came together late on the third day and were not separated until after tea on the fourth, their 199-run partnership the highest for the seventh wicket for India against Australia.
A double century from Cheteshwar Pujara and a century from Wriddhiman Saha helped India take control of the third Test against Australia at Ranchi on Sunday.
Despite not getting any wickets, the Australians were at least able to slow down India's scoring, reducing the prospect of either side winning the match with just four sessions to go and the pitch still favouring the batsmen. India have a healthy lead of 129 heading into the final day's play.
If their dogged partnership was not enough, India piled misery on the Australians who were unsuccessful twice while using referrals - when Saha was on 19 and Pujara on 157.
The express Cummins was by far Australia's most threatening bowler, managing to gain surprise bounce from a slow wicket to claim three of his four wickets with bouncers.
Pujara's undefeated stay at the crease spanned six hours and 52 minutes and he faced 328 balls, hitting 17 boundaries.
After India declared their first innings on 603 for nine, Ravindra Jadeja bowled David Warner and nightwatchman Nathan Lyon cheaply in successive overs to reduce Australia to 23-2.
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These moments allowed Pujara to carry on past 150 in a performance that effectively matched the big hundred made by Australia's captain Steven Smith. Left-arm spinner Steve O'Keefe bowled 77 overs to pick up figures of 3/199.
India number eight, Saha was also going strong at the other end.
Both fell in the final session trying to accelerate with Australia looking pretty jaded by then.
This century will definitely give Saha and the team enormous confidence as the hosts look to take a 2-1 lead in the three-match series before the last Test in Dharamshala.
Luckily for Pujara, he appeared to have got an inside edge though replays did not confirm whether it was bat or pad first. Steven Smith used a review to overturn the original decision of not out.
Cummins, who struck twice in the afternoon session, got Rahane caught behind and Hazlewood bowled Nair with a handsome reverse swinging delivery in the final session.
BOWLING (to-date): Hazlewood 31-9-66-1; Cummins 25-8-59-4; O'Keefe 43-11-117-1; Lyon 29-2-97-0; Maxwell 2-0-4-0.