Since their shock Champions League elimination by Olympique Lyon last week, Pellegrini has been in the spotlight with local media speculating on a possible replacement for next season.
"I'm pleased because we won, we scored three goals but I'm not content with the way we played," the coach told reporters on Saturday.
"We had a bad first half. We need to start with greater intensity and to be more active in the creative part of the team.
"We improved in the second. We need to maintain our winning momentum," added Pellegrini.
Real's only chance of silverware is to win La Liga this year and they started the day level on points with arch-rivals Barcelona but top on goal difference.
The leaders lacked a cutting edge in the first half against a well-organised Sporting and there were whistles from home fans when David Barral opened the scoring for the visitors at the Bernabeu.
However, Rafael van der Vaart, Xabi Alonso and Gonzalo Higuain replied as Real maintained their 100 percent home record in the league.
Director general Jorge Valdano rallied to Pellegrini's defence and gave him the club's full backing again, as he did after the Lyon defeat.
"He is Real Madrid coach and is subject to exaggerated opinions. The club is happy with his work," said Valdano.
"We suffered a blow against Lyon but the team has recovered and you can see his work. Of course he has the backing of the club."
