Top-ranked India finished the day on a strong 319-4 at the Saurashtra Cricket Association ground in Rajkot, supported by the duo's performance.
Pujara made 124 off 206 balls before being dismissed by first-innings centurion Ben Stokes, but owes his ninth Test century and third against England to the Decision Review System.
.@mvj888 celebrates as he brings up his 7th Test ton @Paytm Test Cricket #INDvENG pic.twitter.com/NeTysEAMtD
— BCCI (@BCCI) November 11, 2016
England cricketers wear 'poppies with pride' despite ongoing FIFA spat
The right-handed batsman was ruled lbw on 86 but the TV umpire -- which long-time sceptics India are using for the first time in a home series -- overturned the decision after the ball-tracker showed Zafar Ansari's delivery going over the top of the stumps.
Mr. Pujara's DRS call delights Mrs. Pujara @cheteshwar1 #INDvENG pic.twitter.com/sW8oIYF450
— BCCI (@BCCI) November 11, 2016
Vijay hit 126 during a marathon eight-hour innings and shared 209 runs for the second wicket with Pujara.
Indian skipper Virat Kohli was batting on 26 at stumps while nightwatchman Amit Mishra was dismissed for a duck off the last ball of the day.
The hosts trailed by 218 runs with six wickets in hand in reply to England's 537, their third biggest total on Indian soil thanks to centuries from Joe Root, Moeen Ali and Stokes.
Ton-up Root, unbeaten Ali help England start strong against India
After being stuck on 99 for 10 balls, Pujara reached the three-figure mark with a nervous single as his father and wife applauded from the stands.
Vijay raced to his seventh Test century with two boundaries off Stuart Broad, who is playing his 100th Test, and celebrated it by leaping into the air.
Pujara, 28, flourished in the post-lunch session, twice carting leg-spinner Adil Rashid for fours in the same over.
Rajkot celebrates as @cheteshwar1 hits his 9th Test Century @Paytm Test Cricket #INDvENG pic.twitter.com/5Bs3oZAOEF
— BCCI (@BCCI) November 11, 2016
Vijay was lucky to earn a life on 66 when debutant Haseeb Hameed failed to hold onto his catch off Broad (1-54) in the covers.
It was England's turn to huff and puff in temperatures touching 35 degrees Celsius (95 degree Fahrenheit), and on a surface that had little to offer to the bowlers.
Paceman Chris Woakes impressed the most in the testing conditions, giving away just 39 runs off 23 overs.
Stokes, Ali star as England post commanding total against India
The morning session saw India lose opener Gautam Gambhir (29) to Broad, who struck off his first ball of the day.
.@cheteshwar1 and @mvj888 celebrate on breaching the 50 runs mark @Paytm Test Cricket #INDvENG pic.twitter.com/dZuniImLns
— BCCI (@BCCI) November 11, 2016
The ball hit his pad in front of the middle and off stump, and the decision was upheld after Gambhir referred it to the third umpire.
Before the start of play, England players and support staff observed a minute's silence as a mark of respect to the war dead on Armistice Day.
The players also wore shirts with a commemorative poppy on the right collar.
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