Two wins in their last 12 matches have left Boro six points adrift of the playoff places in the Championship table with three games left to play.
"We want to finish the job we've started and always knew was going to be a long-term one, but we understand that if you don't win football matches the pressure mounts and supporters get irritated and think there's a Jose Mourinho waiting round the corner to come and change everything with a magic wand," Mowbray told Sky Sports News.
"My nature is to keep fighting. This club has shown that sort of resolve over the last 30 years - if we hadn't, where would we be?"
Boro are back in action on Saturday when they travel to Bolton Wanderers.