Real is five points adrift and only two and three points ahead of Celta Vigo and in-form Villarreal after 15 rounds.
Rafael Benitez's team had won five straight, including an 8-0 rout of Malmo in midweek, since the El Clasico defeat to Barcelona, but showed little of that form in the first half with the strike by Soldado off a Cedric Bakambu pass giving the Yellow Submarine a deserved lead.
Read: Barca thrashes Real in El Clasico
Bakambu might have added to the tally before Real woke up in the second half with a string of chances falling to Karim Benzema, who had previously been in a vein of rich scoring form despite his off-field problems.
This time the French star was wasteful with three excellent openings while Cristiano Ronaldo and Gareth Bale also had glimpses of goal but failed to equalize.