Goals from Spanish international midfielder Isco and French striker Karim Benzema either side of half-time were enough to secure the three points, although Depor will feel aggrieved not to have taken anything from the game.
The Gallicians put their more illustrious hosts under pressure for long spells and were unfortunate not to level matters at the beginning of the second period when midfielder Celso Borges cracked a post. Real goalkeeper Ilker Casillas then made a flying save from Oriol Riera moments later.
After a 4-0 derby thrashing at the hands of city rivals Atletico Madrid last weekend, Real manager Carlo Ancelotti will be content to return to winning ways.
However, the Italian will also be concerned at the often scrappy nature of his side's play and the relatively timid performance of star-man Cristiano Ronaldo who failed to score for his third La Liga game running -- the first time he has done so since 2013.
Ronaldo came in for criticism from some Real supporters earlier in the week after holding his 30th birthday party in the wake of last Saturday's derby humiliation.
The World player of the year did set up Benzema for Real's second and struck the bar in the first half, but he was quiet for long periods and missed from two-yards out when presented with a golden opportunity shortly before Isco opened the scoring.
Speaking after the game, Ancelotti said "It was a difficult week and we have come out of it with a victory which is good," in quotes carried by the AFP news agency
"The team was a bit worried at the start, but in the end it went well. The fans have helped us and they understood the moment the team is going through to help us come out of this quickly."
Real and Ronaldo will be hoping for a more convincing performance when they face Schalke in the last 16 of the Champions League on Wednesday.
Saturday's result moves Los Blancos four points clear atop the La Liga table although this will be cut back to one point if second placed Barcelona beats Levante at the Nou Camp Sunday.
English FA Cup
Round five of the FA Cup meant there was no Premier League action this weekend.
There was, however, two all top-flight cup ties Saturday.
West Brom comfortably overcame West Ham 4-0, thanks to a brace from the in-form Brown Ideye and further strikes from James Morrison and Saido Berahino, while Liverpool came from behind to defeat Crystal Palace 2-1 at Selhurst Park.
In Saturday's other cup fixtures, Premier League Stoke City was thrashed 4-1 by second-tier Blackburn.
Meanwhile, Nigeria striker Yakubu scored the winning goal as Reading defeated Derby County in an all Championship affair.
French Ligue 1
An injury hit Paris Saint Germain missed the chance to go top of Ligue 1 after drawing 2-2 at home to Caen.
The reigning French champions had forged ahead thanks to goals from Zlatan Ibrahimovic and Ezequiel Lavezzi but came undone when injuries forced Serge Aurier and Lucas off the park.
Manager Laurent Blanc had used all three substitutions meaning PSG were forced to play the last 10 minutes with nine men.
Although they held on for the majority of that time the Parisians conceded goals in the 89th and then 92nd minute to Emiliano Sala and Hervé Bazile respectively to ensure the spoils were shared.
The result leaves PSG in third place behind Marseille, which drew with Stade Reims on Friday, and table-topping Lyon which heads to FC Lorient Sunday.