Gary Rowett's side took an early numerical advantage in the contest when Bees winger Sergi Canos saw red, with Tom Huddlestone, Cameron Jerome - with his first goal for the club - and Matej Vydra from the spot making sure that it counted.
The Rams claimed another clean sheet to extend a remarkable run that has seen them concede just three goals in their last 14 outings and keep up the promotion pace with Wolves, who are in action at home to Sheffield United in today's late kickoff.
Aston Villa also kept up the pace with a 3-2 home win over basement side Burton Albion in a contest that saw Albert Adomah notch up his 13th goal of the campaign. The Villans have now won their last six league games in a row - their best run in almost 28 years - and are just one point behind Derby in the standings. The Brewers, meanwhile, have now conceded 15 goals in their last four and remain rock bottom of the division, four points from safety.
Cardiff City also kept pace with their promotion rivals with an emphatic 4-1 win at Leeds United, who had Gaetano Berardi sent off on the stroke of half time - their third red in as many games. Neil Warnock's side sit just two points behind Villa in fourth, but do have a game in hand on the sides around them. Leeds have now collected just two points from their last six games but remain 10th in the table and seven points off the top six.
Fulham's ascent up the standings continued as they claimed a comfortable 2-0 home win over Nottingham Forest for their seventh win in eight and fifth in a row. Chelsea loanee Lucas Piazon and Stefan Johansen were on the scoresheet for the Cottagers, who leapfrog Bristol City into fifth place. As for Forest, they drop one place to 16th but maintain a nine-point advantage over the dropzone, although they have now failed to score in six of their last seven games.
At the other end of the table, Hull City's woes worsened as they suffered a 2-1 defeat at Preston North End, Alan Browne securing the decisive goal for Alex Neil's side after Billy Bodin was brought down in the box.
Queens Park Rangers secured a 1-0 victory over Barnsley at Loftus Road courtesy of a curled effort from Josh Scowen early in the second half, holding on for the win despite Ryan Manning receiving his marching orders 20 minutes from time.
Birmingham City did their survival chances a world of good with a 3-1 victory at Sheffield Wednesday, who had Marco Matias and Daniel Pudil both sent off. The Owls did manage to notch up their first goal since December, however, with Lucas Joao pulling one back just before the hour mark.
Up at the Stadium of Light, Sunderland fell to a 2-0 defeat at the hands of Ipswich Town, who had won just one of their previous eight fixtures on the road. Chris Coleman named all three of his deadline-day signings - goalkeeper Lee Camp, midfielder Ovie Ejaria and forward Ashley Fletcher - in his starting XI but the side were a goal down with 35 minutes on the clock when Joe Garner found the net against the run of play. Town soon had a second when Adam Matthews diverted into his own net on the stroke of half time. The result keeps the Black Cats 23rd in the table and three points from the safety line, while Mick McCarthy's charges remain in midtable.
Reading ended the day two points closer to the relegation zone after suffering a 2-0 defeat to Millwall at the Madejski. Both goals came in the space of four second-half minutes, first Leandro Bacuna turning into his own net and then Lee Gregory poking in the second.
Championship results in full: Aston Villa 3-2 Burton Albion, Derby County 3-0 Brentford, Fulham 2-0 Nottingham Forest, Leeds United 1-4 Cardiff City, Norwich City 1-0 Middlesbrough, Preston North End 2-1 Hull City, Queens Park Rangers 1-0 Barnsley, Reading 0-2 Millwall, Sheffield Wednesday 1-3 Birmingham City, Sunderland 0-2 Ipswich Town