Gordon Strachan departed the club 11 days ago after a disastrous start to the npower Championship season, with Mowbray stepping into the vacancy earlier this week.
Mowbray is starting out in his first job since leaving Celtic earlier this year, and has promised fans that he will turn the team into a successful and attractive side to watch, given time.
He told the Middlesbrough Gazette: "It's time for the club and supporters to work together to get us on an upward curve, so I'd ask the fans to be supportive and to be patient.
"Right now it's clear that we are not a free-flowing football team and we will have to get there bit by bit.
"Confidence is a big thing in football. So if things don't always go to plan, I'd ask the supporters to stick with it and keep backing us because their backing really can make a big difference in giving confidence to the team."