Hart allowed Leigh Griffiths to score two late free kicks in the game and the 30-year-old appeared to be at fault for both of the goals.
Southgate said: "Joe's contribution has been high. He has shown great leadership and much more composure before matches.
"The goals were two moments of quality, but we will look at them."
"Composure Joe has been really honest about, that being an area he felt he needed to develop and there is no question he has."