Boro were twice caught on the break in the second half as first Andrea Orlandi then David Lopez struck to fire Brighton back into the play-off places after a wobble.
Boro have slipped from second spot at the turn of the year to 10th and now sit just six points off the drop zone after a run of just eight points from a possible 51 this year.
But Mowbray said he was not considering walking away. "I'm a fighter," he said. "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 have 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."
Brighton boss Gus Poyet was delighted to be back in the play-off positions and praised his side for stepping up a gear after the break.
"We had to inject our own passion," he said. "In the first half the games was very slow and the ground was very quiet and sometimes that can creep into your head and into your game.
"We were comfortable in the first half but we needed more. I told then to go and create their own passion, to make their own tempo and they did that and it made a difference.
"I knew that if we could only get the first goal it would be better for us. With the players we had on the pitch it makes it very hard for Middlesbrough to score a goal."
- Sports & Recreation