Kang-in Lee broke the deadlock in the 70th minute with a precise left-footed finish, and Álex Baena doubled the lead 15 minutes later with a direct free-kick curled into the top corner, sealing a 2-0 win for Atlético at the Metropolitano.
Funes's side had matched the hosts for long spells, pressing with conviction and showing the same daring build-up play that carried them to promotion last season.
"When you don't win you can't leave satisfied," Funes admitted afterwards, but he was quick to praise his players for standing toe-to-toe with one of LaLiga's giants.
"The team came to compete and we were able to go toe-to-toe with Atlético Madrid," he said. Málaga's first-half display in particular echoed the identity that fuelled their promotion campaign, with their pressing, composure in possession and willingness to build from the back keeping them competitive throughout the opening 45 minutes.
The Loja-born coach felt the game had several distinct phases, rejecting the idea that his side had lost their nerve even as Atletico Madrid's superiority told in the closing stages.
"There were many games within the same game," he noted, while also identifying a lack of penetration once his players had broken through Atlético's pressing lines as an area to improve. Substitutions changed the complexion of the match and eventually tipped it Atlético's way, and Funes acknowledged the gap in individual quality between the two squads.
"There's a big difference when you compete against certain teams, because there's a very big gap in many senses," he said. "They put you under that pressure because of the individual quality they have. They showed it today and they'll do it on more days."
One of the match's most debated moments came when an Atlético player narrowly avoided a second yellow card. Funes acknowledged the incident had crossed his mind but declined to use it as an excuse for the defeat.
"It's wrong to think the game turned on that moment, because they didn't end up with a man down," he said, adding that while he believed the challenge merited a card, such moments are simply part of football.
"There will be plenty of incidents that hurt us and others that help us, but the team put in a good performance," he added, steering the focus back to what his team could control. The occasion also carried personal significance for Funes, who was supported by family members at his first match in charge in the Spanish top flight.
"The first time is always very special, and our families didn't want to miss it," he said.
Diego Simeone, whose side needed fresh legs from the bench to break Málaga down, praised both scorers when asked to pick the better of the two goals.
"It would be unfair to Álex or to Kang-in. Both were brilliant goals," the Atlético coach said. "The goalkeeper had just made two great saves. They were two great goals. We needed them. Málaga had a great game. The lads in the first half did a great job so the others could come on afterwards."
On his selection for the opening day, Simeone explained he had simply picked those in the best physical condition to start.
"We understood the best way to approach this game was to play those who were in the best shape," he said. "We had a few situations we could have resolved better. With the substitutions came players with more quality who were able to resolve them."
Asked about Kang-in Lee's application on the pitch, Simeone praised the South Korean's professionalism.
"He's a player who has to work in line with what the game demands. He's a footballer committed to his team," he said.
On Baena, Simeone dismissed any comparison to Antoine Griezmann.
"I'm not looking for someone similar to Griezmann, they're all different," he said, adding that Baena's recent World Cup triumph with Spain was something the squad hoped to draw energy from this season. Asked about the transfer market, Simeone kept his answer open-ended.
"We're always open to something happening," he said, insisting he was not referring to any specific player.