The Serie A side ran riot in a dominant three-goal first half to set up an impressive victory in their debut match against English opposition.
Absent from continental competition since the 1990-1991 season, Atalanta looked right at home in their first Group E encounter as Andrea Masiello netted a deserved opener inside half-an-hour.
Gian Piero Gasperini's side took full control late in the opening half as Alejandro Gomez and Bryan Cristante combined to put the result beyond doubt by half-time.
And the hosts were rarely troubled after the restart as they took early leadership of the group, in the process extending Everton's record of having never beaten Serie A opposition in Italy.
The Toffees have now failed to score in their past three competitive fixtures and face a tough task to repeat their record of having always escaped the competition's round-robin stage.
Ronald Koeman started Maarten Stekelenburg in place of regular goalkeeper Jordan Pickford, but it was opposite number Etrit Berisha who had the busier start as he first palmed away a stinging Gylfi Sigurdsson strike, before Wayne Rooney dinked a delicate chip narrowly over the bar.
Stekelenburg was soon called into action to smother a close-range chance for Masiello, although there was little he could do from the resultant corner as the marauding defender pounced on a favourable deflection to tuck away the opener.
The chances continued to pile up for the hosts and they doubled their advantage in spectacular style before the break, skipper Gomez stepping in from the left and curling a sumptuous strike into the top-right corner from outside the area.
And there was still time for Atalanta to make it 3-0 on the stroke of half-time as Andrea Petagna reeled in a ball over the top and found Cristante running unchecked through the middle for a simple low finish into the bottom-left corner.
Rooney drew a sharp low save from Berisha immediately after the restart as Everton attempted to make amends, though they were fortunate not to fall further behind when Gomez skied a good chance at the other end.
Koeman responded with a double change 25 minutes from time, introducing Sandro Ramirez and Davy Klaassen in place of Rooney and Morgan Schneiderlin.
However, the attacking intent did not yield the desired results as the visitors struggled to create any real openings in the final third and once again drew a blank, meaning they are now winless in five ahead of Sunday's (Monday morning AEST) trip to Old Trafford.
Atalanta will look build on their encouraging start in their next European assignment at Lyon, while Everton host Cypriot outfit Apollon Limassol.
Key Opta stats:
- English sides have lost 10 of their last 14 visits to Italy in European competition (W2 D2 L10).
- Everton have now lost all four of their away games in Italy without scoring a single goal.
- This is only the third time that an English club has lost their opening group stage game of a Europa League campaign. The other two were Birmingham City in 2011-2012 & Manchester United in 2016-2017 – the latter ended up winning the tournament.
- The Toffees conceded three goals in the first half of this game – the first time that they have conceded three times before half-time in a competitive fixture since November 2016 (0-5 vs Chelsea).