The Serie A outfit dominated from the off, scoring all of their goals in the first period through Andrea Masiello, Alejandro Gomez and Bryan Cristante to inflict a third straight defeat on the Blues in all competitions.
Atalanta, who finished fourth in the league last season, took the lead straight from a Gomez corner in the 27th minute, with defender Masiello on hand to guide the ball beyond Maarten Stekelenburg just moments after the Dutch goalkeeper had denied him from close range.
Gomez doubled the home side's lead four minutes from half time with the pick of the evening's goals, gaining possession 25 yards out and curling an unstoppable strike straight into the top corner, putting his side on course for three points.
The hosts put the game beyond Everton just minutes later when an unmarked Andrea Petagna threaded a pass through to Cristante, allowing his teammate to steer the ball beyond Stekelenburg to make it 3-0 at the interval.
Everton saw more of the ball in the second period, but Atalanta remained the better team and came close to a fourth in the 57th minute when a vicious strike from Remo Freuler rattled the crossbar.
The Blues' best chance to pull a goal back came three minutes from the end when Gylfi Sigurdsson's cross was nodded down by Phil Jagielka for substitute Kevin Mirallas to take on the volley, but the Belgian international skewered his shot wide.
Everton are next in action this weekend when they travel to Old Trafford to take on Manchester United in the Premier League, in search of their first win since the opening day.