The deciding moment of the match came just before the break when Iago Aspas was tripped by Diego Llorente inside the penalty area.
Celta's captain took the spot-kick himself and scored, sending Alex Remiro the wrong way.
After the break, La Real dominated much of the possession, with Adnan Januzaj looking their most dangerous threat cutting in from the right.
However, the visitors held on to secure a crucial three points to move seven points clear of Real Mallorca in 18th place.
As for the hosts, they're losing ground in the race for the Champions League spots and are now six points behind Sevilla in fourth.