After a rain delay, Cheteshwar Pujara was run out by debutant Ollie Pope and a downpour followed immediately.
Anderson, who has 544 Test wickets, will have his eyes on equaling Ian Botham's record of 27 five-wicket hauls in the longest format.
However, India batsman Rohit Sharma called on fans to back the team through tough times as these were the same players who helped the side achieve the number one ranking in Tests.
Earlier, India made a decision to play with two spinners in Ravichandran Ashwin and Kuldeep Yadav, whereas Cheteshwar Pujara had a horrific return to Test cricket.
Both Kohli and Rahane survived the early jitters from the English pacers to start the rebuilding work with a 34-run fourth wicket stand before the in-form India captain fell to Chris Woakes.
When play resumed, seamer Chris Woakes, brought in to replace Ben Stokes, who is at a court hearing in Bristol, got the prize wicket of India skipper Virat Kohli for 23 leaving the tourists, who trail 1-0 in the series, at 49 for four.
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While the cloud cover that had aided England's bowlers on day two was absent on a bright morning in St John's Wood, there was still ample movement for the Indian seamers during the morning session.
"I would be so disappointed if I had messed up today because they were ideal conditions to bowl in".
"We exploit those conditions as well as anyone in the world". "For me I find it so much fun when it is like that", said Anderson.
With clouds expected to hang over the iconic cricket ground over the next few days, India may rethink their strategy of playing two spinners. We hardly bowled any bad balls, we didn't give them much to hit, and when you build pressure like that all day, no matter who you are, it is hard. While left-arm spinner Kuldeep Yadav and Ravichandran Ashwin may play a key role to trap Englishmen.
Bairstow and Woakes came together when England were struggling at 131 for five, only 24 ahead of India, with the ball still moving around, but the pair drained the life out of India who found themselves short of pace bowling and over-loaded with spin on a slow pitch.