Warburton joined the club in 2011 as a first-team coach before moving upstairs, where he took overall responsibility for the football side of the club.
But he heads back into the dugout after Rosler left the Bees on Saturday to become Wigan's new boss.
Chairman Cliff Brown said: "We have done very well this season with a squad Mark has helped to assemble and with the existing coaching and back-room staff.
"We believe Mark will offer continuity at a key time and enable us to build on the past two-and-a-half years.
"Mark's coaching credentials are without question, he is highly qualified to deal with both professionals and our young players.
"Mark has a commitment to blooding young players into the first team and to the academy and that remains important for the club's development.
"Mark has done an exceptional job as sporting director since taking that role 30 months ago and we are happy he can step into this new role seamlessly.
"We believe it will help the players and staff to work with someone they know and respect and we look forward to continuing success for Brentford Football Club."
Warburton inherits a squad that are fourth in Sky Bet League One, six points off the automatic promotion places.
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- Brentford Football Club