Spurs dominated large parts of the game yet the match ended goalless despite Manchester United having defender Rafael sent off in the second half.
Speaking to Sky Sports after the game, Dawson admitted his side had felt they could win the match, particularly after the Brazilian’s red card.
“When they went down to 10 men we thought we could go on and win they game but it wasn’t to be,” he said.
“They stood strong, defended well. So we go away with a clean sheet but we’re disappointed not to get the three points.”
Midfielder Luka Modric also revealed he felt his side had played well and that a draw was a fair result.
“The game was very tight with not much space. We played a good game. We are a little bit disappointed. I think a draw is a fair result,” he added.
“That’s a good sign when you’re disappointed by a draw with such a good team. We need to look forward and keep going.
“I think we are doing well, we just need to keep going. Before this game I hoped we could achieve a better result than last season. That would be great for us. So far, so good for us.”
The draw sees United return to the top of the Premier League table, but only on goal difference ahead of neighbours Manchester City. Tottenham, meanwhile, lie fifth, a point behind Chelsea in fourth.