Derry bounced back from their opening day defeat by Finn Harps to dominate affairs with Barry McNamee hitting the side-netting in the first half.
Curtis capped an impressive display by firing low inside the post on the hour.
Cork midfielder Greg Bolger was shown straight red card for a poor challenge on Curtis with 10 minutes remaining.
It was also a first competitive win for new Derry boss Kenny Shiels and it was a deserved victory against a team tipped to challenge for the title.
Patterson saw his close-range free-kick charged down after Cork keeper Mark McNulty lifted the ball in the box from a poor back-pass.
Derry continued to threaten after the break with Conor McCormack sending a long-range effort just over.
Curtis, who set up McNamee for his first-half chance, was Derry's dangerman and the duo combined again for the only goal of the game.
Cork put in a lacklustre performance and Bolger's dismissal completed a miserable night for the Leesiders.
There was applause on 18 minutes in memory of Mark Farren, the former Candystripes striker who died last month.
Farren's number 18 shirt was retired by the club before the game.
A second-half strike from Ronan Curtis earned Derry a first Premier Division victory of the season as they overcame Cork at the Brandywell.