The nine-times European champions dominated the match but, as has happened in recent games, they lacked fluidity in their play and appeared anxious when chances fell to the likes of Gonzalo Higuain, Angel Di Maria or Cristiano Ronaldo.
"We've had eight clear opportunities but we haven't scored," their newly-installed coach Mourinho, who has been on the back foot after Real's dour recent performances, told reporters.
"The only game where we created fewer was in San Sebastian (against Real Sociedad) and we scored twice.
An opening day draw was followed by three consecutive victories with only one goal conceded, but the Bernabeu faithful expect more and they grumbled and whistled at their side during the laboured 3-0 win over Espanyol in midweek.
"We've only conceded once in six matches. We've controlled all the games, haven't had tactical problems and have been defensively solid. We always want to win but if you don't score you don't win," added Mourinho defending their mixed start.
The match pitched the world's richest club by income against a side who only came out of financial administration on Thursday, two years after seeking protection from creditors.
Real spent about 80 million euros on players like Argentina's Di Maria and Germany's Mesut Ozil in the close season, on top of 250 million spent the previous year.
Levante strengthened their squad after winning promotion by landing fringe players from other La Liga sides on loan or with free transfers.
"Each side plays to their strengths. It's clear economic power and football power are different things but we were the only ones who tried to play," said Mourinho.
For Levante the result crowned a special week.
After three straight defeats, in which they conceded 10 goals, they collected their first points back in the top flight with a 1-0 win at Almeria in midweek.
They then secured their financial independence again.
"It proves money doesn't speak in football," Levante president Francisco Catalan told Spanish television.
"The board's principle aim was to take the club out of administration, and we have achieved that. Our next job is to secure our place in the top flight."
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