The former Manchester United and England captain was officially confirmed as the Rams' manager on Saturday (AEDT), having held the post on an interim basis since the departure of Phillip Cocu last November.
But his reign started inauspiciously as Jamie Lindsay's late strike sent the Rams second bottom of the table.
Norwich City, meanwhile, opened up a seven-point lead at the top of the Championship table after a 2-1 win at Cardiff City.
Norwich remains well on course for an immediate return to the Premier League thanks to a fast start in Wales.
Grant Hanley put the Canaries in front after just four minutes, nodding in at the far post after a flick on from a corner.
It was 2-0 after 20 minutes as Todd Cantwell rounded off a slick counter-attack by slotting into an empty net after a smart save from Alex Smithies.
Joe Ralls gave Cardiff hope by thumping into the bottom corner from 12 yards after 65 minutes, but the hosts were reduced to 10 men five minutes later when Marlon Pack was shown a second yellow card and slipped to a fourth successive defeat.
Swansea City remains four points behind Norwich in second thanks to a comfortable 2-0 win at Barnsley.
Ben Cabango opened the scoring just before half-time before Jamal Lowe doubled the Swans advantage with his 10th goal of the season.
Bournemouth missed the chance to go second after a 1-0 loss at home to Luton Town.
The Cherries' task was made more difficult when Jefferson Lerma was shown a straight red card after 26 minutes for catching Tom Lockyer in the face with a stray arm while competing for a header.
Bournemouth pushed to overcome their numerical disadvantage but Luton capitalised as the hosts tired to take the lead after 67 minutes.
Kiernan Dewsbury-Hall curled the winner into the bottom corner from 15 yards.
Watford returned to the play-off places with a 2-0 win over Huddersfield Town.
Tom Cleverley gave Watford a 54th-minute lead by capitalising on indecisive defending from the Terriers.
Alex Vallejo attempted to pass the ball back to Ryan Schofield, who hesitated as Cleverley closed in and beat the goalkeeper to the ball to slide in from six yards.
Joao Pedro made sure of the points 10 minutes later, tapping in at the far post from Kiko Femenia's cross.
Sammy Ameobi struck twice as Nottingham Forest secured back-to-back wins for only the second time this season with a 3-1 victory against Millwall.
Birmingham City manager Aitor Karanka enjoyed a successful return to the Riverside as Scott Hogan struck to earn the Blues a 1-0 win and prevent Middlesbrough from moving into the play-off places.
A 1-1 draw did little to aid Stoke City and Blackburn Rovers's play-off hopes.