The hosts earned victory on day five of the first Test in Ahmedabad to take a 1-0 series lead after winning by nine wickets.
Cook, who spent nine hours and 17 minutes at the crease to score 176, believes that his side put up a fight but that a good start in Mumbai on Friday is crucial.
"There are a number of reasons we didn't win the game - 500 was a good score on that wicket, but our second innings showed there were runs out there," said Cook.
"There was almost a glimmer of hope with five wickets down, Matty [Prior} and I knew what we had to do, but it wasn't meant to be and we have to start better in Mumbai.
"We fought hard, but we just weren't good enough in anything. The pitch got slower if anything. We know how crucial first-innings runs are in the subcontinent, we need to play as close to our potential as possible."
The second Test gets underway in Mumbai on Friday.