Substitute Cole, who had not scored in English football since 2012, struck late on after Premier League Brighton had moved in to a two-goal lead thanks to first-half goals by Anthony Knockaert and Juergen Locadia.
The veteran left back, a seven-times FA Cup winner, joined Derby, managed by his former Chelsea team mate Frank Lampard, in January having left LA Galaxy last year.
Despite the 38-year-old Cole's late effort, Brighton hung on to take their place in the quarter-finals.
"I am pleased for him. He has been working hard for his fitness and we gave him 45 minutes," Lampard said.
"It was not the perfect circumstances but it turned out good for him. He looked calm and the class was evident."
Brighton manager Chris Hughton made eight changes to his line-up with his priority being to make sure his side avoid being sucked into the relegation battle after winning only one of their last 11 Premier League games.
But they still had enough to see off a disappointing Derby side who only came to life in the second half.
"You can always dream," said Hughton, an FA Cup winner as a player with Tottenham Hotspur.
"The reality is it is normally one of the top six that wins the competition; for the likes of ourselves it is about going as far as we can.
"To get to the quarter-finals is a big thing for the club."
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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