Strikers Chris Martin and Conor Sammon both scored twice and England Under-21 midfielder Will Hughes capped another impressive display with a goal to book County's place in the third round.
Derby had lost at home to Shrewsbury and Scunthorpe in the last two seasons, but were far too good for a Brentford team showing 10 changes from the one that beat Walsall on Saturday.
Nigel Clough made only four changes to his side and Sammon had a chance to mark his return to the starting line-up when a long ball reached him on the edge of the area in the second minute, but his shot was charged down and Brentford responded when Jake Reeves fired narrowly over from 20 yards.
But Derby went in front in the 19th minute when Craig Bryson created space on the left and bent in a cross which went in off Martin's shoulder from eight yards.
The Championship side almost had another in the 27th minute when a good move involving Paul Coutts and John Eustace ended with Bryson striking the outside of a post from 20 yards.
Brentford were playing their part in an entertaining tie, but they almost conceded again in the 32nd minute when Hughes set up Sammon, whose shot was acrobatically cleared off the line by Kevin O'Connor.
But the escape was short-lived as Hughes, Martin and Bryson exchanged first-time passes to send Sammon in to slide a low shot under Jack Bonham in the 36th minute.
The game was up two minutes later when on-loan Spurs right-back Adam Smith crossed from the right and Hughes drove a low shot into the bottom left corner from 15 yards.
Brentford, to their credit, made a spirited start to the second half and Reeves had a great chance to pull a goal back, but volleyed over from 12 yards after a free-kick from Sam Saunders was only half cleared.
But Derby were creating numerous chances and they scored a fourth in the 70th minute when Sammon pounced on a rebound eight yards out after Bonham had saved from Bryson when Brentford were caught in possession inside the area.
It was 5-0 seven minutes later when Martin was allowed to evade a challenge and set himself too easily just inside the area and he finished clinically with a low shot past Bonham into the bottom right corner.
Brentford kept going right to the end, but it was Derby's night and Smith almost added a sixth in stoppage time with a right-foot shot that dipped just wide of Bonham's right post.