Sergio Sanchez and Martin Demichelis gave visitors Malaga a surprise early lead with almost identical headers inside the first half-hour but Sami Khedira got Madrid's comeback started with a goal in the 68th minute and Gonzalo Higuain hauled Madrid level two minutes later.
After relentless pressure, Karim Benzema slid home the winner in the 78th.
"The team didn't start the game as it should have," said Madrid coach Jose Mourinho. "The second half was completely the opposite. The team wanted to clean out the trash it had made in the first half. The attitude, the concentration of the team changed." Malaga coach Manuel Pellegrini, Mourinho's predecessor at Madrid, was disappointed at his side's collapse.
"Giving goals away to Madrid on their own pitch makes it very difficult," he said. "The tie is still open, but we deserved more." Madrid could meet archrival Barcelona in the quarterfinals if both teams win their last 16 ties. Barcelona plays its first leg at home to Osasuna on Wednesday.
There were surprise results in three other last 16 fixtures on Tuesday. Third-tier Mirandes claimed a 2-0 home win against topflight Racing Santander, second-tier Alcorcon had a 2-1 home victory against top division club Levante, and third-tier Albacete drew 0-0 with Athletic Bilbao.
The return legs will all be next week.
Madrid looked like adding its name to the list of unexpected scorelines after a sluggish first-half performance against Malaga.
Under coach Jose Mourinho, Madrid had won its past three matches against Malaga by an aggregate of 15-1.
Madrid selected a strong starting lineup with Cristiano Ronaldo, Kaka and Jose Callejon supporting Gonzalo Higuain in attack.
Malaga coach Manuel Pellegrini selected another former Madrid employee in Ruud van Nistelrooy as his lone striker.
Malaga's Santi Cazorla had the first chance when he shot straight at Iker Casillas from the edge of the box in the fifth minute.
The warning was not heeded and Malaga took the lead soon after when Sergio Sanchez headed Cazorla's corner past a helpless Casillas in the 10th.
Again, Madrid failed to learn the lesson and fell further behind from the same route. This time, Cazorla's corner was met by Martin Demichelis, who beat Casillas to the ball and headed into the net in the 29th.
Madrid continued to enjoy the better of the possession, but looked rusty after the winter break and slow in defense when Malaga counterattacked.
Mourinho used all of his substitutions at halftime, replacing Kaka, Alvaro Arbeloa and Jose Callejon with Sami Khedira, Benzema and Mesut Oezil.
"I said to the players at halftime that it was a shame I couldn't make 11 substitutions because I would have changed all of them. I wasn't singling anybody out," said Mourinho. "I prefer to go home with 10 men and lose but have the feeling that I took a risk and tried to change something. You have got to take risks." The changes settled quickly and Khedira broke into the box to shoot past Malaga goalkeeper Willy Caballero and give Madrid hope in the 68th.
Just two minutes later, Higuain pounced on a misdirected backpass, rounded Caballero and rolled into the net from a tight angle to pull Madrid level.
"The truth is we had to wipe out what we did in the first half," said Higuain. "We came out in the second half with a different motivation and we are winning, which is the important thing." The atmosphere inside the Bernabeu changed immediately after the equalizer. From jeers of frustration, the crowd turned to cheer every Madrid pass.
The 10-minute turnaround was completed when Higuain produced a deft flick for Benzema to run behind the defense and slot into the corner for the winner.
"The players turned things round with a lot of merit and a lot of character," Mourinho acknowledged, before questioning whether some of his charges had overindulged during the break for Christmas. "There are people who interpret holidays as time to have fun, to travel, to eat with their dad, uncle, aunt, grandmother ... and perhaps arrive back here a little different," he said.
