Late goals from substitutes Obafemi Martins and Peter Lovenkrands sealed a dramatic come-from-behind victory for the hosts in a full-blooded Tyne-Tees derby.
Newcastle's first home win for months lifts them out of the bottom three at the expense of Hull City and gives Alan Shearer's side renewed hope of retaining their top flight status.
But it's a different story for Middlesbrough, whose 11th straight away defeat means they remain three points adrift of safety with just two games remaining, the first at home to Aston Villa on Saturday before they travel to West Ham on Sunday week.
And they will be without record signing Afonso Alves who was stretchered off in the first half after suffering a suspected broken foot.
Boro took an early lead through an Habib Beye own goal, but Steven Taylor equalised almost immediately to leave the second-bottom club needing back-to-back wins to have any hope of avoiding the drop into the Championship.
The last time Middlesbrough achieved such a feat was August, and Southgate admitted: "We have to lift ourselves even though it's out of our hands and we're relying on other results.
"I still believe I'm the right man to take this club forward and I won't be walking away from it. We're still in it. Only when it's mathematically impossible will we down tools but we're definitely not giving up the fight."
Shearer's first victory in his sixth game as interim manager gives Newcastle hope of prolonging their 16-year stay in the Premier League, but the 38-year-old refused to take credit for his inspired substitutions.
He said: "It's a magnificent feeling to come out on top in such an important game. The players were brilliant and I asked them to show courage and determination and the ability they've got.
"It would have been easy to feel sorry for themselves after going a goal down but there were some big performances from some big men.
Newcastle host Fulham on Saturday before travelling to Villa on the final weekend and Shearer added: "I've asked them for three huge performances and that's the first of three. It's a first step towards safety.
"We've got to go and do it again against Fulham when it will be just as tough.
