The win ended Los Rojiblancos' four-match losing run in La Liga.
Aguero netted the game's only goal following a swift counter-attack to ease the pressure on Atletico coach Quique Sanchez Flores.
Zaragoza threw everything at their visitors in the closing stages and struck the bar twice in the final two minutes through substitutes Ikechukwu Uche and Nicolas Bertolo, but there was no way through.
The result moved the visitors up four places to seventh in La Liga, while Zaragoza - who had picked up 10 points from the previous 12 on offer at home - stayed 17th.
Zaragoza will consider themselves unfortunate not to have gone a goal up in the 65th minute after a frantic scramble in the Atletico six-yard box, but the visitors not only cleared their lines but went up the other end and nicked the winner.
With the home defence stretched, Diego Forlan switched play to Aguero on the right and the Argentinian worked space for himself well before lashing a fierce shot past Franco from a tight angle.