Tuncay Sanli struck with five minutes remaining to hand Boro a 2-1 success over the Premier League new boys after Stewart Downing had missed a penalty for the home side.
Southgate told Sky Sports: "It was a hard-fought victory, and we made hard work of it. We should have had it wrapped up long before we actually did.
"We need to be more ruthless in both boxes. The penalty was a bit of a turning point. At 2-0 they would have been dead and buried.
"The pleasing thing is that we kept our nerve and Tuncay came up with a great finish to give us the win."
On Faye's dismissal, Southgate said: "It was a bad tackle. I thought the referee got the decision right."