India posted 436 replying to England's first-innings total of 246 for a sizeable first-innings lead of 190 at Rajiv Gandhi International Stadium.
England rode player-of-the-match Ollie Pope's epic 196 to post 420, setting India a tricky target of 231 on a turning track.
Debutant spinner Tom Hartley claimed 7-62 to bowl out India for 202 and help England to go 1-0 up in the five-match series.
"It was always going to be challenging to chase 230 on a fourth-day wicket which was turning," Dravid said after India's loss in an extended final session on the penultimate day.
"I thought we probably had 70 runs less on the board in the first innings when conditions were pretty good to bat in on day two."
Yashasvi Jaiswal (80), KL Rahul (86) and Ravindra Jadeja (87) missed out on hundreds while Axar Patel and Srikar Bharat fell short of fifties.
"I thought in those kinds of situations we got ourselves into some good starts but we did not truly capitalise," former India captain Dravid said.
"So in some ways, we left those 70-80 runs back in the first innings... I'm a little bit more disappointed with our first innings."
While Pope provided a masterclass in how to play spin using his feet and frequently playing the sweep and scoop shots, several India batters looked ill at ease against the turning ball.
Dravid attributed it to their lack of first-class cricket exposure.
"A lot of these guys play a lot of white-ball cricket and don't get a lot of time to play first-class cricket. But they are learning, they are getting there."
Dravid heaped praise on Pope's stellar knock, which he felt made all the difference.
"It's one of those games where you... come out and give your best and if somebody does something exceptional, you shake his hand and congratulate him," Dravid said. "And for me that is important."
The second test in Visakhapatnam begins on Friday.
