Tottenham dominated the first half, but Jermain Defoe, Rafael van der Vaart, Aaron Lennon and Gareth Bale were all guilty of spurning good chances.
Their profligacy looked as if it was going to go unpunished when Benoit Assou-Ekotto smashed home a deflected half-volley 16 minutes from full-time, but James Morrison popped up in injury time to snatch a point for West Brom after a scramble in the box.
Villas-Boas, who replaced Harry Redknapp at the helm this summer, has now gained one point from his two opening matches.
The 34-year-old insists there is still plenty of reason to be optimistic about the season ahead, though.
"It is very, very important to get that first win but there are still 36 games ahead of us," Villas-Boas said.
"We have one more point than we did this time last year.
"The players have been excellent. They have been receptive and there is a good feeling about what we are doing. We are just waiting for the results to accompany us a little bit more."
Spurs were on top throughout the first half, but the only difficult save Ben Foster had to make before Spurs opened the scoring was from a Assou-Ekotto free-kick just after the break.
Van der Vaart looked a shadow of the player that has terrorised Barclays Premier League defences for the last two seasons and despite looking sharp on occasions, Defoe struggled to make an impact on his own up front.
Emmanuel Adebayor caused problems when he came off the bench in the second half, but Villas-Boas admits his team as a whole must be sharper in front of goal next week when they welcome Norwich to White Hart Lane.
"We have been working well during the week, trying to prepare and get the result," Villas-Boas added.
"Now we just have to be a bit more aggressive in front of goal. We have to finish the chances we have.
"It was frustrating not to get something from the first half today. We created lots of opportunities and we were extremely attacking.
"We could have got something from the first half but we didn't so it was a pity we conceded that goal right at the end."
Tottenham's start is in complete contrast to that of West Brom, who now have four points on the board after today's draw and their win over Liverpool last weekend.
Their new manager Steve Clarke inspired an improved performance from his team in the second half after giving his side a rollicking at the break, and he also made a key impact on the game by bringing on Romelu Lukaku for Shane Long on the hour.
The Belgian teenager, snubbed by Villas-Boas at Chelsea last season, used his hulking frame to give William Gallas a torrid time.
"We are pleased with Romelu," Clarke said. "It's great that you can look to your bench and see people who can change a game and Romelu is one of them. He did a good job for me, but so did Shane Long. He had the hardest part of the game."
Clarke also reserved praise for Morrison, who remained calm to stab home from 10 yards 30 seconds into injury time.
It was something of a surprise that the former Middlesbrough man was even on the pitch, though, as he had rolled his ankle in the second minute of the game.
"We weren't sure whether he would continue," Clarke said. "It did cross my mind to bring him off a couple of times, but you have to leave someone like James on the pitch because if you leave him on he can pop up with a goal and he did that.
"I never thought it wouldn't be our day today. I always thought we'd get another chance to score and we did."
