Boro went into the game with injury problems in defence and were caught cold in the 19th minute by a clever free-kick from Ben Davies which was swept into the net from 10 yards by Michael Jacobs.
Stuart Parnaby wasted a good chance to equalise when he fired high over the crossbar after a Lukas Jutkiewicz shot was blocked and Derby increased their lead shortly after Jonathan Woodgate limped out of the match in the 36th minute.
A poor clearance by Rhys Williams in the 42nd minute allowed Jeff Hendrick to dispatch a swerving right-foot shot from the edge of the area into the top left-hand corner of Boro's net and Davies rattled the crossbar with a free-kick just before the half-time whistle.
Conor Sammon had a shot blocked by Jason Steele and his follow-up header cleared off the line in the 47th minute but he sealed victory in the 65th minute when he pounced on the rebound after Steele had saved from Jacobs.
Jutkiewicz pulled one back when he turned in Ishmael Miller's low cross in stoppage-time but it was too late to save Boro from a fourth defeat in their last five away games.
"It's disappointing to lose any game but you have to give them credit," Mowbray said. "They are a team full of energy and some very good players. We never got to the races at any stage of the game.
"We've done okay so far this season but for whatever reason, we didn't get going at all. But we will get stronger as the season goes on and players are back from injury.
"There was probably almost a team missing but I don't want to use that as an excuse. Derby played well and good luck to them."