Spurs were leading through Jon Grounds' own goal in the 26th minute and Boro posed almost no threat going forward.
"We were too 'end of season' in the first half," said Southgate. "The goal we conceded epitomised that. Not acceptable. In the second half we responded brilliantly and were more positive and we asked Tottenham questions."
Grounds put through his own goal in the 26th minute after Dimitar Berbatov's impish piece of skill set up an opening.
The Bulgaria striker, with his back to goal, flicked the ball off the floor and volleyed a cross to the far post. Aaron Lennon met the cross and his finish came off Grounds and went in.
The equaliser came 21 minutes from full-time when Stewart Downing's long-range effort deflected in. After drawing with Manchester United and narrowly losing to Chelsea, Boro can now look upwards rather than towards the relegation zone.
"I'm pleased we have shown we are not dead in games and we can create enough, even away from home, to win matches," said Southgate.
"But I'm not pleased with the first half because it was not acceptable for the supporters who travelled. I'm glad we got ourselves out of jail in the second half.
"We couldn't have been a lot worse. I felt it was important. We had a few pats on the back recently with the way we've played but the first half was not us, not the players I know, and it didn't represent the talent we have in the dressing room or the professionalism we have got.
"Hopefully it is a lesson learned because we are a young group and if they thought they could come here and just play a nice game of football and win, we've learned the lesson very quickly that you cannot do that in this league."
In the end, Afonso Alves or substitute Tuncay Sanli could have snatched victory for the visitors.
"Possibly we created enough to win it but you do not deserve to win it if you play like we did in the first half," Southgate added.
Spurs head coach Juande Ramos admitted his side are struggling to motivate themselves now European qualification has been secured and there is no danger of falling from mid-table.
"Possibly the lack of importance of the points at the moment meant the players dropped their guard a little," he said. "The game is 90 minutes and we have to look at the game as a whole."
However, Ramos' focus will not be on which players he wants to retain.
"All the players have had their opportunities, they have all been used at some point when I've made substitutions and they've all had their chances," he said.
"We always want to produce the best performance possible but you have to understand the situation the team is in.
"We are in the middle of the table, the points aren't excessively important - we can't reach the top part of the table and we don't seem to be in danger.
"It is not easy to get the maximum motivation when the team is in this situation."