Kohli was unbeaten on 91 and Pujara remained 97 not out as India reached 210 for two at tea after winning the toss and opting to bat in the first ever Test match in the port city of Visakhapatnam.
A stray dog ran onto the field and held up play for a few minutes, forcing a slightly early tea with Pujara three runs away from his 10th test hundred and second of the series.
On a surface expected to assist the spinners from early on, Kohli had no hesitation in batting but his plans suffered an early jolt.
Fit-again opener Lokesh Rahul, who replaced Gautam Gambhir in the Indian side, lasted just five deliveries as fast bowler Stuart Broad dismissed him in his first over.
Rahul fell for a duck, hanging his bat away from his body to edge Broad to Ben Stokes at the third slip.
His opening partner Murali Vijay looked in good touch, hitting four crisply driven boundaries in his 20 before he fell to James Anderson.
England's most successful wicket-taker bowled cross seam to extract steep bounce and hit Vijay on his glove for Stokes to complete a simple catch.
The wicket marked a successful return for Anderson, who had not played since August with a shoulder injury and replaced fast bowler Chris Woakes.
The hosts were 22-2 at that stage when Kohli and Pujara joined hands to rebuild the innings.
The right-handed pair had little problems in negotiating the English bowlers and their only moments of concerns in the first session were couple of run-out opportunities.
Kohli was then dropped at fine leg by Adil Rashid off Stokes with the batsman mistiming a pull shot while batting on 56.
Pujara also survived a England review for leg before against Moeen Ali with replays showing he was hit outside the line.
Cook tried everything to separate the duo, including asking his bowlers to bowl with eight fielders on one side to stem the flow of runs and frustrate the batsmen.
India handed a Test debut to off-spinner Jayant Yadav, who replaced leg-spinner Amit Mishra, a decision prompted by the presence of seven left-handers in the England batting line-up.
Original source: Kohli, Pujara punish England