The Celtic captain says the squad owe it to the manager to turn their form around.
He also urged supporters to get behind the team and the manager following the reigning champions' defeats by Ross County and Aberdeen.
"We now need to do it for him," said the Scotland midfielder. "We've got to take responsibility."
Brown said the players were "devastated" by the 2-1 defeat by Aberdeen, which reduced Celtic's lead at the top of the Scottish Premiership to three points.
He remains confident, though, that the squad has the ability and mentality to prevail in the title race.
"Stick with us," Brown said in a message to the Celtic fans.
"We're disappointed with the last two games, but we'll bounce back.
"We've got two huge games coming up in the league at home and we'll need their support.
"The manager's been good since he came in, he's put 100% into the job and a lot behind us as well.
"He's coping well, he's always positive. He's always looking at how we can improve as a team and players. He's bouncing through it.
"We speak all the time, we go through a lot of things together."
Brown said there were sticky patches under previous manager Gordon Strachan.
"We've got to take a lot of responsibility," he said. "It's us who are out on the park.
"We're still top of the league and that's what people have got to remember.
"We had tough times under Gordon and we came through that, we became better players and went on to win trophies.
"That's what we aim to do at the end of this season.
"We're still three points clear. We know what it's about, we've been there and done it before and we'll do it again this year.
"Winning leagues is always hard, losing two games in a row is always hard, but it's about how you bounce back. We've got the mentality in that dressing room and the belief that we can do it."
Scott Brown says that it is time for the Celtic players to repay the faith Ronny Deila has shown in them.