Mido gave Boro the lead only for a second half brace from Jermain Defoe to secure maximum points for Harry Redknapp's side.
"We've played really well in the first half and looked in control of it. The work we've done in the week paid off, we got the goal and looked strong," Southgate said after Saturday's game.
"In the second half we invited pressure onto us and we didn't keep the ball long enough and pose them enough questions going forward, and if you do that there's a danger of what happened happening.
"The home team are bound to have a spell in the game and you have to see it through. Ross Turnbull has made several great saves and the turning point is the first goal.
"We've got ourselves in a very good position twice and we should have more points on the board.
"They're out on their feet and have given absolutely everything and it's tough for everybody to take.
"But we have to learn from it. We have to keep possession and keep going forward and not invite pressure."