Wednesday bounced back from defeat at Millwall on Saturday to record their third win from the opening four games thanks to Kadeem Harris's simple finish, nine minutes into the second-half at Hillsborough.
Former Barcelona midfielder Cocu was given an indication of how tough it will be to take the Rams back to the Premier League when they fell behind 2-0 at half-time to goals from Andreas Weimann and Josh Brownhill.
Jack Marriott gave Derby late hope, but it was not enough to salvage Cocu's unbeaten start.
Middlesbrough eased the pressure on new boss Jonathan Woodgate as the former England defender got his first win in charge with a 1-0 victory over Wigan, thanks to Britt Assombalonga's first-half goal.
Birmingham were 2-0 winners over Barnsley, while Blackburn ground out a 1-0 victory away to Hull.