Having seen Bournemouth defeat Watford by the same scoreline on Friday, the pressure was on Steve McClaren's men to respond in South Wales.
Chris Martin missed an early penalty for the visitors, but they took the lead midway through the first half thanks to Scott Malone's own goal.
Martin atoned for his earlier miss with a goal in first-half stoppage time - his 16th of another productive league campaign.
Victory saw Derby move back level on 54 points with Bournemouth, while Middlesbrough are a point further behind in third after Grant Leadbitter's penalty earned them a 1-0 success at fellow high-flyers Brentford.
At the opposite end of the table, bottom club Blackpool got just their fourth win of the season with a 1-0 victory over Brighton and Hove Albion, thanks to a goal from Jamie O'Hara.
Struggling Wigan Athletic earned an impressive point from a 0-0 draw at Ipswich Town, while a late goal from Ricardo Fuller saw Millwall heap further pressure on Nottingham Forest manager Stuart Pearce.
In the West Yorkshire derby, Leeds United's 2-1 win at Huddersfield Town was overshadowed by a serious injury to Huddersfield's Tommy Smith.
The defender had to be airlifted to hospital after suffering a head injury and was taken by air ambulance to Leeds General Infirmary.
Wolves needed a last-gasp goal from James Henry to rescue a 2-2 draw at Bolton Wanderers, while Reading defeated Sheffield Wednesday 2-0 and a late equaliser from substitute Danny Ward saw Rotherham United earn a 1-1 draw at Charlton Athletic.
Elsewhere, Blackburn Rovers overcame Fulham 2-1, and the game between Birmingham City and Norwich City ended goalless.
