Jose Luis Morales put Levante ahead in the seventh minute after a defensive slip from Raphael Varane while Roger Marti doubled the lead in the 13th from the penalty spot after a handball by Varane, with the spot-kick awarded after consultation with the video assistant referee (VAR).
Lopetegui brought on Gareth Bale on at half-time, while Karim Benzema and Dani Ceballos soon followed.
The striker's curling 20-yard strike hit the post in the closing stages, and that was the nearest Madrid came to an equaliser, with Mariano's late tap-in ruled out for offside in a match that kept the officials busy.
Real have now gone five games without a win in all competitions, losing to Levante, Alaves, CSKA Moscow and Sevilla, as well as drawing with city rivals Atletico.
A trip to Barcelona is next for El Clasico after a midweek Champions League visit from Viktoria Plzen.
Madrid had 34 shots against Levante and Olazabal was pivotal in seeing out the result by making 11 saves - the most a goalkeeper has made in a single La Liga game this season. We need to change this dynamic.
A four-time Champions League victor, Madrid have come to rely on Varane.
Real Madrid are really struggling right now.
It is their worst run since May 2009, when Juande Ramos was sacked.
The hosts thought they had got an immediate foothold back into the clash when Marco Asensio, the last Real player to score, nodded in the rebound after Sergio Ramos' header had had bounced off the crossbar. However, Levante's form has dipped in recent matches, as they have won just three of their last eight games under Lopez.