Jason Puncheon skied an early penalty before Cameron Jerome, Mile Jedinak and Marouane Chamakh all came close to finding the net as Palace caused their hosts numerous problems at White Hart Lane, particularly in the opening 45 minutes.
Second-half goals from Christian Eriksen and Jermain Defoe eventually put the game to bed, but Pulis feels that it could have been a different story had one of their efforts gone in
"We played so well first half and really the game should have been over with the opportunities, chances and the way we played," he told Sky Sports.
"Coming in at half-time, you just don't know how you're not winning a game by a couple of goals. But against a team like Tottenham, a side with that much quality, they're always going to have their time in the game.
"In the second half, they played well. The first goal was disappointing, it's a long ball down the middle of the pitch that we don't deal with. There's the flick-on and it's a two-pass goal.
"And the second one, on another occasion, I think Defoe could be ruled offside. That was disappointing.
"Even in the last 20 minutes, their keeper has made a great save, Chamakh's had a great header off a corner which he should score and we've had really good opportunities. We've just got to take those opportunities."
However, Pulis said he was proud of the showing from his men and believes that he has seen enough in recent weeks to be positive about their chances of survival.
"The lads have got to take a lot of credit for the way we took the game to Tottenham in the first half," he said.
"We played well at Chelsea and we played well at Man City, we've looked very, very solid and we've got good pace in the team and good movement and we like to play through the pitch quickly.
"Whether it's small passes or balls up the side, it was something like 17 or 18 crosses we did today.
"The breaks are not going for us at the moment but what we've got to do is maintain the spirit, commitment and belief we showed today."
The defeat means that Palace slip to the foot of the Premier League table and Pulis acknowledged the need to bring in reinforcements during the January transfer window.
"We've got to add a little bit of depth to the squad we've got, I think we go bottom tonight but the results have dragged a lot more teams into it. With 17 games to go, it's a good challenge," he said.
"We need to bring a few players in and we need to strengthen certain areas to give us a chance because at times, when we're up against it, we need different options.
"So we have to grind a few results out, not only play well but also grind a couple of results out.
"We know who we want but we've obviously got to work at a certain level at Palace.
"There's not a lot of players in the pool that we can work with and get. We understand that and the lads who are here, as you've seen today, have done fantastic.
"It's a juggling act and something that we're trying to get and make sure that if we do it, we get it right. It's always difficult."