Di Maria was left on the bench by Madrid coach Carlo Ancelotti for the second time in a row but came on for Luka Modric with 12 minutes remaining and immediately caused havoc in the visitors' defence.
The 26-year-old winger, who is unsettled following the summer signing of James Rodriguez and may still leave the club this summer, was not directly involved as the Colombian opened the scoring but gave life to a previously sterile Madrid attack and the goal arrived just three minutes after his introduction.
"The game changed with Di Maria," Simeone told reporters afterwards. "It's logical - he's the best player they have."
Asked if he really believes his compatriot to be Madrid's finest footballer, the Atleti coach replied: "He's the player most likely to create openings in the opposition half."
Madrid's lead lasted just six minutes, however, as Raul Garcia levelled from a corner late in the game.
"Thanks to the goal we have a 50 per cent chance in the home game on Friday," Simeone remarked. "It's 50-50 because Madrid always score goals and thinking about a 0-0 draw won't be good.
"We set up the game to play to our characteristics and the match was more how we wanted it than how they wanted it."
Summer signing Mario Mandzukic missed a couple of decent chances for Atleti but Simeone was pleased with the former Bayern Munich man's overall contribution.
"He did some great work for the team defensively," the Argentine said. "We want Mandzukic and Raul [Jimenez] to block the counterattacks. The team was compact, it worked fairly well."