"He has signed a five-year contract."
Atletico did not give any financial details regarding the transfer fee but the Spanish media have estimated that Sevilla paid Palermo 12 million euros.
Vazquez arrived at Palermo in January 2012 and played 73 Serie A games for the Sicilian club in the last two seasons, scoring 18 goals and setting up 17 more.
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The Argentina-born player has already had a stint in Spain, playing on loan at La Liga side Rayo Vallecano from Palermo in the 2012-13 campaign, scoring three goals in 18 games.
Vazquez made his Italy debut in March 2015 but was not selected in the country's Euro 2016 squad.
Nicknamed 'El Mudo', Vazquez is replacing Eder Banega following the Argentinian's move to Inter Milan earlier this month. He is Sevilla's eighth signing of the summer.
Sevilla, under new coach Jorge Sampaoli, will take on Champions League winners Real Madrid in the UEFA Super Cup on Aug. 9 in Trondheim, Norway.
(Reporting by Cindy Garcia; Editing by Amlan Chakraborty)
