Stifled by Atletico in six winless meetings last season, Messi shredded its vaunted defense with run after run, passing for Neymar to score in the 12th minute and for Suarez to double the lead at Camp Nou in the 35th.
Messi was guilty of also setting up Atletico's goal in the 57th when he fouled Jesus Gamez in the area, allowing Mario Mandzukic to score from the spot.
Barcelona took control of second place, one point behind leader Real Madrid. Atletico in third trails the pacesetters by four. Madrid still has a game in hand.
Elsewhere, Sevilla tightened the race at the top with a 2-0 win at Almeria thanks to goals by Vicente Iborra and Jorge "Coke" Andujar, moving into fourth place two points behind Atletico.
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Elche moved off last place with a 2-1 win at Athletic Bilbao in its first victorious visit to the Basque club after 20 trips.
Eight months ago Atletico won the league title in a winner-takes-all finale at Camp Nou, whose fans ended up honoring the victors while jeering their own team.
On Sunday, it was "Messi! Messi!" that rang out after Suarez's goal.
Barcelona has had one of its most tumultuous weeks in recent memory after it lost its appeal of a one-year transfer ban for violating FIFA's rules for registering youth players.
That blow was followed by the firing of its sports director on Monday and president Josep Bartomeu announcing the club would hold early elections after this season.