Jude Bellingham opened the scoring after 19 minutes with a superb solo effort, powering through the Malaga defence before firing past goalkeeper Alfonso Herrero.
Bellingham was involved again seven minutes later when Herrero saved Kylian Mbappe's effort. The England midfielder headed the rebound towards goal and the ball eventually deflected into the net off the Malaga goalkeeper.
Mbappe then made it 3-0 in the 30th minute, producing an excellent volley from Trent Alexander-Arnold's cross for his fourth league goal of the season.
Mourinho made five changes to his starting XI, including handing Alexander-Arnold and Marc Cucurella their first starts of the campaign, but Madrid quickly found their rhythm.
The tempo dropped after the break, although Bellingham struck the post and Madrid remained comfortable.
Malaga were awarded a penalty in the 85th minute, but the decision was overturned following a pitchside review after the ball struck Yan Diomande's arm without intent.
Madrid then punished Malaga's attacking ambition as substitute Arda Guler combined with Vinicius Junior before finishing in added time.
The win gives Madrid nine points from three matches, two ahead of Atletico Madrid. Barcelona, on six points, can reduce the gap when they host Rayo Vallecano on Monday.
Bellingham, who received a standing ovation when substituted, said he was enjoying the freedom Mourinho had given him.
"I've got the freedom to go higher or lower if we need, I'm enjoying myself a lot," Bellingham told Real Madrid TV of his attacking midfield role under Mourinho.
"It's an honour to play for this club and in front of these fans and the ovation was very nice."
Mourinho was also pleased with Madrid's first clean sheet of the season.
"Football isn't just about having fun with the beauty of the goals - it's also about going home thinking that we're a solid unit," he said.
Madrid next face Real Betis away on Friday before beginning their Champions League campaign against Inter Milan.
