Jon Grounds put into his own net to hand us the lead as we had the better of the first half and Dimitar Berbatov struck the post on the hour.
However, Stewart Downing's deflected equaliser changed the game and Southgate's men carved out the best of the latter chances, Afonso Alves going agonisingly close inside the last 10 minutes.
Boro have now picked up points at Villa, Chelsea and the Lane in recent weeks and pushed United all the way last weekend.
"You can't start games in the way we did," said Southgate. "In the end, we created enough chances to win it but you don't deserve to win games with the way we played in that first half.
"We were sloppy and too far off them and Spurs are a team that if you allow them to play football they are as capable as anyone in this league of doing so.
"The first half wasn't acceptable for the supporters who travelled. We've 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."
Gareth Southgate was as delighted with his side's second half reaction as he was disappointed by their first half display at the Lane.