The Rams were ahead just 24 seconds after kickoff when Matej Vydra found the bottom corner from 25 yards out.
David Nugent then added a second four minutes into the second half, slotting home from Vydra's through-ball.
Rowett told the official Derby site after the game: "We scored at the right times and at crucial times today. We also managed to navigate ourselves through some periods when we knew that we wouldn't have the ball.
"There were some moments where I felt we could have done better and the save from Scott Carson was a really key moment in the game. These matches hinge on moments such as those and he is so experienced and made a fabulous save.
"We kept our shape when we needed to and as the game we went on I felt we got stronger."
Derby's first win in five Championship games sees them lift the Brian Clough Trophy and move up to 13th in the table, one place above their rivals.