Atletico were 2-1 up from the first leg and looked in cruise control as Diego Costa and Radamel Falcao both struck early on.
Jesus Navas pulled one back, and then Ivan Rakitic struck in injury time, but it was too little too late for Sevilla.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.