Atletico claimed bragging rights over city rivals Real Madrid thanks to Mandzukic, whose early strike helped the Liga champions to a 1-0 second-leg victory and 2-1 success on aggregate.
Croatia international Mandzukic - a high-profile recruit from Bundesliga champions Bayern Munich - netted the winner in the 81st second at the Vicente Calderon, the fastest goal in Supercopa history.
Atletico midfielder Koke said it was pleasing to start the 2014-15 campaign on a winning note, having lost leading scorer Diego Costa, Filipe Luis, Thibaut Courtois and David Villa during the close-season.
"It seemed that everything was becoming more difficult with team-mates leaving but we trusted in the core of the group and the new players who have come in with excitement," Koke said.
"Now, we have started with a trophy at home with our fans."
Koke insisted Friday's triumph over Real was not revenge for May's UEFA Champions League final defeat to their Madrid neighbours.
"It's not that it's revenge, we won the league and they won the Champions League," the 22-year-old added.
"Now we have the Supercopa. We are very proud of the team and the people."
Antoine Griezmann, who made the move to Madrid from La Liga rivals Real Sociedad, also lauded the performance of fellow new recruit Mandzukic.
"Our intention is to win every match," the Frenchman said post-game.
"I already knew what the crowd was like and it is impressive. Mandzukic was 10 out of 10."
Meanwhile, Atletico captain Gabi said the Supercopa was crucial to the club's attempts to win back-to-back league titles.
"It's an important title and we continue to grow," Gabi said.
"Over two legs we can compete with anyone, in a league it is more difficult. It's very important for us."
