The hosts reached 162 for one at lunch after losing Gautam Gambhir but still trail England by 375 runs. Vijay was unbeaten on 57 while local boy Pujara was 62 not out at the interval.
The touring side, who posted 537 in their first innings of the five-match series, enjoyed an early breakthrough after India had resumed on 63-0 when paceman Stuart Broad trapped Gambhir leg before with his first delivery of the morning.
The left-handed opener missed a straight delivery from Broad, who is playing his 100th Test, and was out for 29 to end a solid first-wicket stand of 68.
That was the hghlight of the session for the English bowlers on a placid wicket, however, as Pujara joined Vijay at the crease and soon found his rhythm with some crisply driven boundaries.
The 28-year-old Pujara, considered a specialist Test batsman, thrived against England's spin bowlers but needed all of his defensive guile to negotiate a hostile spell from Chris Woakes when the fast bowler hit him three times on the helmet.
Vijay looked solid at the other end and brought up his fifty with his second six against left-arm spinner Zafar Ansari but survived some nervous moments against leg-spinner Adil Rashid in the last over before lunch.
Pujara followed Vijay in reaching his half-century with an off-driven four against Ben Stokes, who was wayward and conceded six boundaries in his four overs.
The duo added 94 for their unbroken second-wicket stand.
Original source: Vijay, Pujara lead India response to big England total