Dawson (66 not out) and Rashid (60) shared a record 108-run eight-wicket partnership to frustrate the Indian bowlers on what looks like a good batting pitch.
India were 60 for no loss at stumps with Lokesh Rahul (30) and Parthiv Patel (28) at the crease. The hosts still trail visitors by 417 runs in the first innings.
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England, who started the day on 284-4, were reduced to 321-7 in the morning session after overnight centurion Moeen Ali departed for 146 but Dawson and Rashid kept the visitors going.
The duo's partnership bettered England's previous best eighth-wicket stand in India of 97 by Douglas Jardine and Hedley Verity on their first-ever tour in 1933-34 in Chennai.
The England innings ended 45 minutes into the evening session with Amit Mishra cleaning up Jake Ball for 12.
Ali century makes up for poor England start
Allrounder Dawson, who was a late inclusion in the England squad before the fourth Test, showed his batting prowess during his 148-ball stay, hitting 5 fours and a six.
Dawson's effort was the highest by an England No. 8 on debut in Tests, surpassing David Bairstow's 59 against India in 1979.
Rashid, who recorded his second Test fifty in his 10th game, also looked solid during his gutsy effort as he fell one short of his Test best of 61.
Earlier, Ali seemed relentless in his drive to consolidate England's position after adding 26 runs to his overnight score but his efforts were cut short just after the first drinks interval of the day.
Indian skipper Virat Kohli's ploy of getting his pacemen to bowl short at Ali paid off as the left-hander mistimed a pull on one of Umesh Yadav's rising deliveries.
Yadav and Ishant Sharma took two wickets each while Ravindra Jadeja, with three wickets on the opening day, stood out on the bowling list for India.
The hosts have already pocketed the series 3-0.
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