Thanks to the second consecutive half-century opening partnership between Strauss (38 not out) and Cook (29 not out), England have now taken an overall 147-run lead over the hosts with 10 wickets in hand.
India, needing early wickets to come back into the match, were thwarted by the England openers who had no problems in handling Indian seamers Irfan Pathan and S Sreesanth.
Strauss got into his groove early and executed a superb cover drive off Pathan, prompting captain Rahul Dravid to bring in Anil Kumble in the seventh over of the innings. Kumble's first delivery turned sharply but it beat not just the batsman but also the wicketkeeper to go for four byes.
Despite Kumble getting some turn from the rough patches bowling round the wicket to the left handers, both the batsmen played cautiously working the ball in the gaps and kept the scoreboard ticking with ones and twos and the occasional boundary.
Cook, after his first innings of 60 runs, looked composed and slowly after getting his eye in, showed the range of shots he can play with a back foot cover drive off Kumble being the best of them all.
Earlier in the morning, the Indian innings ended at 323 with Sreesanth being adjudged leg before wicket off Matthew Hoggard for one.
For England, it was Hoggard who turned the table by bowling incisively to cut through the top order of the Indian attack and return with figures of six for 57 in 30.5 overs.