England reached 158-1 at tea on the opening day after being put in to bat, thanks to an unfinished 106-run stand for the second wicket between left-handed opener Strauss and Shah who was unbeaten on 50 at the break on Saturday.
The hosts lead 1-0 in the series following their nine-wicket win in the second Test at Mohali. The first match ended in a draw at Nagpur.
India captain Rahul Dravid, playing his 100th Test, elected to field after winning the toss but the hosts were only able to take one wicket in two sessions.
India went into the match with three seamers for the first time in the series, but Irfan Pathan, Shanthakumaran Sreesanth and Munaf Patel were unable to put pressure on England even on a pitch that initially offered bounce.
England lost Ian Bell (18) in the first session but not before the makeshift opener had given his team a 52-run start. He batted patiently before falling to a loose drive, caught by Harbhajan Singh at point off Sreesanth.
Bell opened the innings when in-form Alastair Cook was ruled out of the game due to a stomach ailment, just a day after strike bowler Steve Harmison had pulled out due to a shin injury.
But Strauss and Shah denied India further success with their contrasting knocks in hot and humid conditions.
Strauss was all patience as he waited for loose deliveries to punish, while Shah exploded with a flurry of boundaries after the lunch-break.
The debutant marked Pathan for punishment, once flicking and twice driving him straight for three fours in an over. He reached his half-century in style, driving off-spinner Harbhajan through the covers for his eighth four.
Strauss continued to bat confidently and reached his first half-century of the series, and the sixth of his career, when he pulled Patel for his sixth four.
He has so far hit one six off leg-spinner Anil Kumble and nine fours in his 165-ball knock.
England made three changes from the side that played at Mohali, bringing in Shah, off-spinner Shaun Udal and paceman James Anderson.
The hosts replaced leg-spinner Piyush Chawla with seamer Sreesanth.
Star batsman Sachin Tendulkar made his 132nd Test appearance, an Indian record.