Mauro Icardi scored a brace in Monday's 3-0 win away to Lazio, with Marcelo Brozovic also on target at a wet Stadio Olimpico.
Inter moved within six points of leaders Juventus, who are seeking an eighth straight league title as their domestic dominance looks set to continue.
And Spalletti, who has stated Inter's aim is to qualify for the Champions League, will not get carried away with his side's fine form.
"The performance is the fundamental thing," Spalletti said to Sky Sport Italia.
"We have the possibility to control games the way we did in the first half, whereas in the second we gave the ball away too much, allowed Lazio to cause us problems and in those situations I see a lot of what we should not do.
"This is why I say we're not the anti-Juve, because if we make those errors, we are anti-nobody. We had one decent half, that's it. Let's not get ahead of ourselves.
"Anyone who wants to leave things the way they are had better get out of the way. The president [Steven Zhang] has spoken clearly. These are the ambitions of Inter, to have this attitude and control the game at the Stadio Olimpico against a very impressive Lazio.
"The results might not always be like this, but the approach has to be for Inter."
Spalletti opted to leave Stefan de Vrij on the bench, the centre-back having conceded a penalty against Inter on the final weekend of last season to contribute to the Nerazzurri qualifying for the Champions League in place of Lazio.
"I didn't think it was fair to play De Vrij when he would be jeered by the entire stadium, which is what would have happened," Spalletti said.
"He is a remarkably sensitive lad and if he had made a couple of errors, he could have gone into a spiral."
Captain Icardi, who now has six goals in four Serie A appearances, agrees with his coach that Inter should not be considered Juve's main Scudetto rivals.
"We are not the anti-Juventus, as we are a few steps away from what they have done in recent years," the Argentina attacker told Sky Sport Italia. "But we are happy with what we are doing. We are on the right track and want to continue.
"The team is improving, we are working to improve and play the ball, which we managed to do very well today. We found good spaces and the moves to score the goals, reflecting the work we do in training every day."