The Championship club were beaten 2-0 at the Riverside Stadium as they exited the competition in the fifth round.
"We have to get back to business now," Mowbray told reporters. "We have to start winning again. Our only concern this year is to try to get out of league.
"The top two looks beyond us now but we've got every chance of finishing in top six. We've got a 12-game season to achieve that goal."
Seventh-placed Boro host leaders Cardiff City in the league on Saturday.