Chelsea remain in sixth place in the table on 48 points from 31 games, while Everton move up to seventh with 46 points from the same number of matches and are making a big push for European football next season.
Beto scored his first on 33 minutes after receiving a superb pass from James Garner, who celebrated his first England call-up this week with an outstanding personal display, before firing a second just past the hour mark through the legs of Chelsea goalkeeper Robert Sanchez.
Iliman Ndiaye added a superb third with 14 minutes left to play as he curled the ball into the top corner for Everton’s biggest victory over Chelsea since 1987, and the first time they have recorded back-to-back wins at their new stadium on the banks of the River Mersey.
Chelsea have now lost four games in a row in all competitions and have had a difficult few weeks in which they were also dumped out of the Champions League by Paris Saint-Germain.
"The responsibility and accountability is with me," Rosenior told Sky Sports. "We set up the team a little differently today, concentrating on a 4-4-2 shape and I thought we were comfortable (before Everton scored).
"The difference in the game was in both boxes, they (Everton) were clinical and it was a wonderful finish from Ndiaye."
Rosenior was unhappy with the way his side conceded the first two goals and said it had been a worrying pattern of late.
"Goals give you energy and at the moment we are gifting goals away," he said. "We started the game edgy and then we played our way into it.
"But it seems to be happening too often that we give cheap goals away and that gives energy to the opposition and the crowd, and their confidence goes up.
"Everton fully deserved the victory and it is something we have to change very quickly."
Garner now has six Premier League assists this season, having managed only four in his entire career before the campaign, form that has been rewarded with an England call-up.
"I’m so happy, I’m over the moon. It’s a dream," he said. "He (England manager Thomas Tuchel) rang me and said I’ll be involved; it’s been a crazy week, and to top it off with a win is even better."
