Alan Stubbs' free-kick sealed the win to make it two wins from two, while Spurs have suffered consecutive defeats to dent their hopes of a top-four challenge.
Moyes said: "We got fourth three seasons ago. We're going to do the best we can but I'm not going to come out and say we are going to do that.
"We don't have that expectation and I don't need to burden the players or myself with it. But I'll probably make myself think that I should."
Dimitar Berbatov struck the post in the second half and Jermain Defoe forced a stunning save from Tim Howard, but Spurs' £40million strikeforce could not find a way back into the game.
"I really thought we played well, the effort and workrate was incredible," said Moyes.
Mikel Arteta also impressed and was the architect for two of the goals.
"He's got the other side of his game now, maybe the spell at Rangers helped him," Moyes added.
"I hope he could make the Spain team, I think he would be in the England team if he was English."
Joseph Yobo picked up an early groin strain but wanted to carry on.
"Hopefully we instil the right thing in players, and they showed that in abundance," Moyes added.
Spurs boss Martin Jol admitted the only positive from the game was his team's efforts to get back in it.
"I felt we were a bit unlucky, if we got to 3-2 it may have been a different story," said the Dutchman.
Jol now has Younes Kaboul on the sidelines after the defender picked up a hamstring injury.
"We have 10 defenders at the club and we've used all them that have been fit," Jol added.
"We are still waiting for one or two players to come back and we will do better when they come back.
"You could throw in the towel after three goals but they came out and worked hard, that was the positive thing."