The pair were each the subject of disciplinary proceedings for actions in the game, which saw Real Madrid beat city rivals Atletico 4-1 after extra-time.
Atletico coach Simeone ran onto the pitch after Cristiano Ronaldo struck the fourth from the spot and was involved in a heated exchanged with Real defender Raphael Varane.
The Frenchman kicked a ball in the direction of Atletico's coaching staff, with Simeone sent from the dugout by referee Bjorn Kuipers shortly after the confrontation.
Simeone will miss the next game in UEFA competition while Alonso, suspended for the final in Lisbon, is out of the Super Cup final after leaving his position in the stands to celebrate with his team-mates on the touchline.
The Spaniard picked up a yellow card in the semi-final against Bayern Munich, which kept him out of the final at the Estadio da Luz.
Both clubs have also been fined for improper conduct of their players, due to picking up five or more yellow cards in the match, and of their fans for setting off fireworks.
Atletico must pay UEFA €21,000 while Real have been hit with an €18,000 fine.
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