Real coach Mourinho threw caution to the wind with a triple substitution after delivering a withering half-time team talk with his side trailing 2-0, calling them "rubbish."
Two of his replacements, Sami Khedira and Karim Benzema, and Gonzalo Higuain led the fightback, netting between the 68th and 78th minutes to turn the tie and spare Real's blushes.
"I told the players at half-time I would have changed all 11 if I could have," Mourinho told reporters.
"I had considered making changes before the break but I don't like to do this. I asked them to clean up the rubbish of the first half."
Argentina's Higuain pounced on a wayward back-pass to level two minutes later and Benzema slotted the third after a wonderful back-heel from Higuain as Malaga desperately tried to keep the floodgates closed.
"We gave them a goal at a crucial moment," Malaga coach Manuel Pellegrini said referring to Sanchez's under-hit back-pass for the equaliser.
"We should have come away with a better result. We had the chances to finish them off."
Pablo Infante was the hero for Mirandes, scoring after 32 minutes and laying on the second for Mikel Martins just after the restart, leaving La Liga's lowest scoring team Racing with a tough task in the second leg.
Last season's losing finalists Barcelona are at home to Osasuna on Wednesday. The second legs will be played next week.
