England were 74-4 when bad light ended play on the final day of the opening match in a three-test series.
Such a near-miss was extraordinary as Pakistan collapsed from 102-3 at tea to 173 all out 21 overs later.
Pakistan had batted first, skipper Misbah ul-Haq declaring on 523-8 before England responded with 598-9, Cook's innings of 263 the third-longest in test cricket.
Cook said his team's resurgence made him just as proud as any of his test wins as England captain.
"The fact that you concede 520 and then you're the side who are pretty much going to win the game, an inexperienced side too," said Cook. "In the next 10 days of test cricket we have to show the same kind of resilience and character."
Cook said he gave England little chance of winning when Pakistan reached tea with seven wickets still in hand.
"It just shows in these conditions things can change very quickly," the 30-year-old said.
England's Adil Rashid, whose bowling figures of 0-163 in the first innings were the worst by a debutant in test history, was the architect of Pakistan's collapse as he finished with 5-64, his five wickets coming in the space of 53 balls.
"He's had a tough introduction, bowling on probably the worst wicket you'd ever want to make your debut on as a leg-spinner," said Cook.
"Full credit to him for really having that character and confidence in his ability to not get too down on himself."
Pakistan's Misbah, who made a second-innings half-century, cited Mohammad Hafeez's foolish run out on 34 and his own dismissal - bowled by Moeen Ali as he skipped forward for a slog - as crucial errors that nearly cost his team dear, but praised his bowlers for denying England victory.
"We were looking to take wickets early on, bowl the spinners when the batsmen are not set and then maybe at the end we had the option of bowling with the seamers," Misbah told reporters. "That plan worked."
Dubai will host the second test from October 22, with the series concluding in Sharjah (November 1-5).
