The Toffees, who are now just seven points of 18th-placed Stoke, were 4-0 down after just 37 minutes in what was a woeful first-half showing.
Things did improve slightly after half-time and Calvert-Lewin managed to score a 64th minute consolation goal - but the Merseyside club went on to lose the game 5-1 and have now won just one of their last nine games.
Sam Allardyce was clearly far from happy after the game, and the former England boss slammed his side's 'crap' performance.
"Any tactics and any system you pick is worthless if the players don't play to their best," he was quoted as saying by Sky Sports
"None of our first XI in the first half played anywhere near their best and that is why we were tumbling goals left, right and centre with our pathetic performance.
"None of it was any good. We got better in the second half but far too late...all the team played crap.
"I was angry. I'm still angry. I think accepting an instruction is part of a player's responsibility at the level required. We tried to give the most instruction in the time we had available.
"There were clear instructions today because there was a good performance by Swansea against Arsenal [on Tuesday] that gave them a good victory and we based our plan on that," Allardyce revealed.
"Unfortunately, Swansea played at the top of their game and we played at the bottom of our game. If you play at the Emirates, you've got to play at your best and our 11 didn't play anywhere near our best, that's why we were tumbling goals in a pathetic performance."