India now trail 1-0 in the five-match Test series after a 31-run loss at Edgbaston.
Virat Kohli and co will look for early wickets on Saturday as the second England-India Test match enters its third day.
But it was new-ball partner Mohammed Shami who struck first when, from around the wicket, he had Keaton Jennings, aiming across the line, lbw for 11.
Rahane, who did all the hard work to survive 44 deliveries, nicked Anderson's straight delivery to first slip Alastair Cook. England's record run-scorer made a fluent start to his innings but was beaten all ends up by Ishant Sharma and nicked behind.
- Promising Pope -The exit of England's all-time leading Test run-scorer meant 20-year-old debutant batsman Ollie Pope walked out with the hosts in trouble at 32 for two.
The tourists were buoyed by Root's dismissal but any suggestion of a batting collapse was dismissed in the afternoon session as Bairstow and Woakes combined brilliantly.
How to see the Perseid meteor shower peak this weekend
And, with NASA meteor expert Bill Cooke dubbing the Perseids "the best shower of the year", this certainly isn't one to miss. That's when the earth will have turned into the meteor stream, which means the show will be just above the horizon line.
The first was the result of an absolutely stunning away swinger that bowled Murali Vijay off the fifth ball of the day.
It was Shami - once again - that prevented further damage as he took advantage of Buttler's tendency to walk across his stumps to pick up another LBW.
England have yet to decide whether Chris Woakes, a fellow pace-bowling all-rounder, or off-spinner Moeen Ali will replace Stokes.
Having come together shortly after lunch with England struggling at 131-5, Woakes (120 not out) and Jonny Bairstow (93) shared a match-changing stand of 189.
Team India will look to bounce back strongly on Day 3 of the second Test against England at Lord's.
Finally, Kohli (23) who had battled hard for 70 minutes and 57 balls, forced into committing a mistake by Woakes.
Danielle Wyatt, England's woman cricketer, who is active on social media praised him for running out Pujara on his first day on the field.