Second-half goals on the break from Andrea Orlandi and David Lopez made it eight points from a possible 51 for Boro since the turn of the year.
And with a cushion of just six points on the bottom three Mowbray's side are not completely safe from the drop with four games left.
But the defiant manager said: "I'm a fighter. I'm not someone who gives in. It is not something I have considered.
"We know the situation we are in isn't great. We know this recent run of results has not been good enough. Everyone at the club is bitterly disappointed about the situation.
"But my nature is to dig in. It is my job to put things right. Ultimately you have to ask the chairman about my situation, but he is from Teesside like me and I am sure it is in his nature to dig in too."
Of the game he said: "If anything I thought we were the better side in the first half.
"But we gambled in the second half and pushed more men forward and we got punished."