Lee Carsley's first game in charge of Brentford ended in defeat as Michael Morrison's second-half goal and Clayton Donaldson's late strike earned Birmingham City a 2-0 win at Griffin Park this evening.
Austrian forward Marco Djuricin spurned two good chances for the home side inside the opening 20 minutes, firing wide when well placed on both occasions.
Button then did well to keep out Morrison's close-range header shortly before the break, though the offside flag had already been raised.
Once things were underway, Donaldson almost broke the deadlock when he cut inside and curled an effort beyond Button's far post.
Gleeson and Demarai Gray both tested their aim from range as the half wore on, but couldn't trouble Button, before the visitors did finally edge ahead through Morrison.
The defender rose highest to head home Cotterill's corner on 71 minutes for his first goal for the club.
Carsley - who made who 53 appearances for his hometown club Birmingham City as a player - saw Alan Judge hit the bar for his side in the final minutes, but Gary Rowett's men wrapped up the win when Donaldson struck deep into added-on time.