Defending champions Manchester United now trail leader Arsenal by nine points.
"We're not as close as we'd like to be," Moyes said.
Ninth-place Tottenham is lagging 10 points behind the Gunners and will rue its missed chances.
"It was a good response to a big defeat to Manchester City," said Villas-Boas, referring to the 6-0 loss at City last weekend that put his future in doubt. "It was not perfect, but it was a good response from a group of players that wanted to do well. The result kept us in touch with a group of teams at the top."
Villas-Boas lashed out at journalists for a perceived lack of respect in the aftermath of his team's drubbing.
"A couple of articles attacked my integrity and my achievements with the club," Villas-Boas said.
United was again without the injured Robin van Persie while Nemanja Vidic started after recovering from a head injury.
Villas-Boas made three changes from the Spurs side that lost to City, with Vlad Chiriches, Mousa Dembele and Nacer Chadli replacing Younes Kaboul, Lewis Holtby and Erik Lamela.
Looking for its first home win against United in 12 years, Tottenham got off to a strong start and dominated possession in an entertaining first half.
United keeper David De Gea boxed a corner from Walker in the first minute and the visitors were kept in their own half by strong pressing from Tottenham.
United left back Patrice Evra was lucky to escape a booking when he fouled Aaron Lennon, with referee Mike Dean playing the advantage.
Spurs' first clear chance came when Tom Cleverley clumsily lost the ball before Lennon who was excellent throughout the match down the right flank threatened De Gea with a low shot.
United responded with a move ignited by Wayne Rooney, who stole the ball from Chiriches before serving up Antonio Valencia. But the England forward's attempt at goal from Valencia's resulting cross was blocked by Sandro.
Tottenham was logically rewarded for its dominance after a clumsy challenge from Jonny Evans led to a free kick that Walker converted from 20 yards (meters) with a low shot drilled under the jumping United wall in the 18th minute.
Tottenham continued to press as a painfully slow United team struggled to string passes together. Spurs should have scored in the 28th after Roberto Soldado started a brilliant move with Paulinho, crossing the ball for the Brazilian who played it back for Soldado behind Evans. But the Spanish striker, alone in front of goal, fired his shot over.
Tottenham was made to pay for its missed chances after an error by Walker, who flicked a cross back into Rooney's path. The United forward made the most of the golden chance and scored from close range with a clean finish in the 32nd, prompting chants of "Rooney, Rooney" in the away stand.
The second half started slowly until Sandro turned Cleverley with a fine dribble and silenced the United fans with a superb swerving shot into the top right corner in the 54th.
The lead hardly lasted as Spurs 'keeper Hugo Lloris fouled Welbeck in the box two minutes later. Rooney slammed the penalty past the Frenchman to equalize and move above Robbie Fowler into 5th place in the all-time Premier League top scorers, with his 164th goal.
"It is not the first time that I have seen us concede penalties against Manchester United where players leave their leg to make contact and then fall down," Villas-Boas claimed. "I can understand it is difficult for referee Mike Dean to analyze but I don't think he is in a good position to judge and for me it is a bad call."
Lennon received a standing ovation as he came off to be replaced by Andros Townsend before Soldado was substituted by Jermaine Defoe, but Villas-Boas' attempts to shake United's defense did not pay off.
Townsend failed to connect with Walker's precise cross in the 79th minute at the end of a counterattack.
"Tottenham had their own pride, which they showed today in their performance," said Moyes, admitting the result was disappointing.
"I think the big games are all to come in the second half of the season ... There will be a lot of teams now looking to chase them (Arsenal) down and hopefully we will be one of them."
