India are left dangling in a dangerous position with another 156 runs still needed in order to save the follow-on.
Having scored a classy hundred at Lord's in the second Test, Rahane again came to the fore with an unbeaten 52 even as some of his illustrious teammates threw their wickets away after getting starts.
Rahane added 74 for the fifth wicket with Rohit Sharma (28), who looked in good nick before committing a harakiri when he decided to step-out to Moeen Ali's part-time off-break only to provide Stuart Broad at mid-off with simplest of catches.
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This was after Virat Kohli (39) showed glimpses of returning back to form but his nemesis Jimmy Anderson bowled full swinging delivery which he nibbled to Alastair Cook at first slip.
Skipper Mahendra Singh Dhoni was batting on two alongside Rahane when the umpires called for tea.
The post-lunch session started with Kohli and Rahane looking to build a partnership. But they could only go as far as 48 runs before India lost another wicket.
This time it was the big one of Kohli, who had until then, fought hard with patience and determination before Anderson accounted for Kohli.
Rohit then walked to the crease, getting his first chance in this series. He started well, rotating the strike with Rahane, as Moeen Ali (1-37) was deployed from one end. Rohit used his feet well against the part-time spinner while Rahane was watchful yet scoring at a decent pace.
They got the scoreboard ticking again and soon celebrated their 50-run partnership, the mark coming off only 78 balls in the 61st over of the innings.
Two overs later Rahane celebrated his half-century, another sterling effort after the hundred at Lord's.
He reached there off 92 balls, with five fours. The 200-run mark for India also came up in the same over.
But then, Rohit played a careless stroke in a bid to keep on attacking Ali, throwing caution to the wind as tea-time approached, and holed out to Broad at mid-off.
Earlier, Broad (2-34) bowled a brilliant spell in the morning session to reduce the visitors to 88/3 at one stage.
Anderson opened proceedings on the day but it was Chris Woakes (0-10) who was deployed alongside him from the other end. It was a clear ploy by England to keep attacking with either Anderson or Broad (2/23) from one end, with Woakes and Chris Jordan (0-23) playing second fiddle.
This tactic worked as Anderson first worked up the batsmen with his swing early on and then Broad charged in to create more trouble.
There wasn't too much in the pitch but the two bowlers still got the odd delivery to create doubts in the mind of the batsmen and that is how both the crucial wickets came about after the 50-run mark had come up in the 22nd over.
Pujara was the first to go, looking to leave a short delivery from Broad in the 24th over, as the ball followed him and kissed his glove on its way to the debutant keeper Jos Buttler who snared his first victim in Test cricket. He hit three boundaries in his 52-ball knock.
Albeit unlucky, it was another start thrown away on tour by the prolific Saurashtra batsman.
Kohli came out to bat thereafter, under pressure from his recent few failures, but got going despite some nervy moments at the crease.
He quickly added 32 runs for the third wicket with Vijay at the other end, but then Broad struck once again. He was troubling the opener outside his off stump and created enough doubts in his mind to make him play-and-leave at one delivery that took an edge and rattled back to his stumps.
Rahane then joined Kohli at the crease and they saw off the session, adding 20 runs for the fourth wicket and taking India past the 100-run mark in the 35th over.
There was only a minor hiccup in the 37th over when Moeen Ali (0-13) almost had Rahane dismissed caught as the ball appeared to touch his gloves but umpire Rod Tucker failed to notice it.