Boro have now won three games in a row to climb into the play-off positions, while this defeat sees Brentford lose their second consecutive game.
Middlesbrough deserved the opener when it came just after the half-hour mark. Leadbitter picked the ball up 25 yards out and unleashed a thunderous strike past Button.
Leadbitter almost doubled his tally and the lead moments later, sending his effort just over the crossbar.
It was all Middlesbrough again as the second half began and it did not take long for the second goal to come.
With an impressive forward move, Nsue broke down the right to deliver a pin-point cross which was met by the sliding Albert Adomah.
It was Adomah again who was at the heart of the action moments later. The Ghana international was on the end of another fine Nsue delivering, but fired his shot into the side-netting on this occasion.
Nsue was involved again as Boro took a 3-0 lead. Bamford headed home after another cross from the Spaniard.
The hosts were in a winning mood and did not seem put off when Dean Whitehead was given a straight red card in the 83rd minute for a two-footed challenge.
Six minutes later, Kike completed the rout with a deft chip over Button to make it 4-0.