Wayne Rooney twice cancelled out Spurs leads, firstly after Kyle Walker had blasted in a low free-kick and then after a sumptuous strike from Sandro.
Villas-Boas felt United were handed a "very, very unfair" spot-kick in the second half, after Danny Welbeck went down under a challenge from goalkeeper Hugo Lloris, that gave them a second equaliser.
But he was overwhelmingly encouraged by Tottenham's response to last weekend's 6-0 defeat to Manchester City, a response that began with a midweek Europa League win against Tromso.
Villas-Boas said on BT Sport: "We had chances to win the game. I think we probably deserved to win the game and were a bit unlucky in the first half not to go 2-0 up.
"We had good intensity over 90 minutes. I think it was a very, very unfair penalty.
"To respond in this fashion against a team that put five past the second team in Germany (Bayer Leverkusen) was good for us. We showed great strength of character and played extremely well.
"We are extremely happy with the performance but not so much the result, because a win would have taken us above them, but it's a good response, not a bad result."
Villas-Boas has come under growing pressure to deliver positive results as his summer signings bed into the team.
Defeat at the Etihad Stadium ramped up that pressure, and led to talk of Villas-Boas fighting for his job.
"You can only look in the table to see the Premier League champions (United) are just one point from us," he said.
"You see there is obviously an agenda that goes against Tottenham's results.
"We have to do better.
"We have to bounce back and I think we did that in a great fashion today.
"There's great unity in the team to bounce back. When we do we'll be ever so close to the top again."
Pointing out Tottenham are through to the League Cup quarter-finals and the knock-out stage of the Europa League, he added: "At the moment I think there should be some respect."
Rooney called the outcome "probably a fair result", although it left United nine points behind pace-setters Arsenal in the Barclays Premier League.
"We're on a good run in the minute but we need to turn some of those drawns into wins," Rooney said.
"Today it was a difficult game, Tottenham away. A point is probably a good result, a good point.
"But it's the other games where we need to turn those one points into three and it would make a massive difference for us."
Rooney described Sandro's 25-yard shot into the top right corner as "a great strike".
Tottenham defender Kyle Walker defended the quality in the home team's ranks, and insisted Villas-Boas is the right man to lead the team forward.
Walker said the intention was to "show the fans something and prove to ourselves we are a top team".
"We've got a lot of quality players in there," England defender Walker said.
"The team spirit in the changing room is first class.
"The results haven't gone the way we've wanted to but we've stayed strong as a team and mentally and physically, training has been top class and it showed today.
"It might take a bit of time but you can see the players we've brought in are class, they're top class international players.
"We'll be up there. I've every faith in the team and in the gaffer as well that we can turn this around."
Asked if all it required was patience to see success delivered, Walker said: "I guarantee it."