India needed a little less than one session to seal the win, with Australia adding just nine to their overnight score as they were bowled out for 241 before India rattled off 50 quick runs.
Clarke made 130 in what had looked a decent first innings total of 380 for Australia, but India took the game away from them with their captain MS Dhoni making a mammoth 224 and Virat Kohli 107 in India's reply of 572.
An unbeaten 81 from Australia's debutant Moises Henriques highlighted their second innings but Ravichandran Ashwin took five for 95 to set up India's victory.
"Full credit has to go to India," Clarke said. "They played outstandingly, especially Dhoni. That innings changed the game completely. And Ashwin's bowling.
"Moises's debut is a positive, but we have got to turn things around. Work hard in the nets and get ready for the next Test match. The pitch looked a lot worse than it played. Second innings, it played tougher, but that's what you expect in the subcontinent. No excuses, we got outplayed."
Dhoni's innings was marked by his attacking intent, with six maximums and 24 fours, but he was happy to credit his teammates with setting the stage after a partnership of 93 from Sachin Tendulkar and Cheteshwar Pujara and another worth 91 between Tendulkar and Kohli softened up the bowlers.
"It was important to play some shots," he said. "The first [significant] partnership was important. Sachin and Pujara batted for a period of time. Virat and Sachin was crucial. It tired the bowlers a bit. Those partnerships were crucial.
"Ashwin bowled brilliantly. In the second innings, he was well supported by Bhajji [Harbhajan Singh]. Ashwin stuck with his off-spinners, which was good. He was willing to bowl in one area, and that's a big positive. It's important to be focused now, but at the same time it is important that the bowlers take some time off now."
The second of four Tests begins on Saturday in Hyderabad.
Follow us @SprtNationalUAE