Nicolas Pareja's first-half free-kick saw Sevilla edge Celta de Vigo 1-0, extending their unbeaten league run to four matches.
Unai Emery's side are into fourth – five points behind Barcelona and Atletico.
Griezmann continued where he left off before the mid-season break by scoring twice in Atletico's 3-1 win over Levante.
And the forward was once again instrumental on Saturday against Lucas Alcaraz's side in La Liga's opening match of the year at the Vicente Calderon.
Griezmann broke the deadlock in the 18th minute with a towering header from Guilherme Siqueira's cross and doubled the tally shortly after the restart by just nodding over the line after Diego Marino kept out Mario Mandzukic's initial effort.
Substitute Nabil El Zhar's first league goal of the season just after the hour handed Levante a lifeline but Diego Godin's header eight minutes from time ended hopes of an unlikely comeback.
Atleti's victory adds to the feel-good factor at the Calderon following the announcement earlier this week that Fernando Torres, who took a watching brief from the stands, is to re-join the club.
Diego Simeone's side move level on points with second-placed Barcelona and are now just one point adrift of leaders and city rivals Real Madrid, although Carlo Ancelotti's side have two games in hand.
Elsewhere, Deportivo La Coruna climbed three points clear of the relegation zone – beating Athletic Bilbao 1-0 thanks to Ivan Cavaleiro's first-half goal.
