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Playing without suspended manager Diego Simeone after his red card in their 3-2 opening defeat at Liverpool, Atletico took charge from the start.
Atletico opened the scoring after four minutes through Giacomo Raspadori, who capitalised on a loose ball in the box. Robin Le Normand extended Atletico's lead in the 32nd minute, firing home the rebound after Griezmann's back-heeled effort from a corner was saved.
Griezmann eventually reached his milestone with a strike from close range in first-half stoppage time, celebrating with an Atletico shirt bearing "200" on the back as the club's record goalscorer.
Frankfurt threatened a comeback when Jonathan Burkhardt's deflected effort wrongfooted keeper Jan Oblak in the 57th minute but Giuliano Simeone restored a three-goal cushion with a clever near-post header from a corner.
Alvarez completed the rout in the 82nd minute, chipping home a penalty after VAR spotted a handball by Robin Koch.
