First-half goals from Megan Finnigan and Jess Park put the visitors 2-0 up at Anfield before Hanna Bennison rounded off their victory.
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With 27,584 watching on and Liverpool's stunning win over champions Chelsea last weekend, the stage was set for Matt Beard's side to put on a show for their fans.
But Everton spoiled their Anfield party with a resounding win which also saw them claim their first points of the 2022/23 season.
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Finnigan put Brian Sorensen's side ahead as she headed in a loose ball from close range.
Park made it 2-0 with a fine piece of skill to round Liverpool goalkeeper Rachael Laws before slotting the ball home on 87 minutes to silence the home faithful.
It rounded off another intriguing weekend in the WSL, the pick of Sunday's matches seeing Chelsea get the better of title rivals Manchester City.
Manchester United won 2-0 at West Ham, while Aston Villa beat Leicester by the scoreline.
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There was also a win for Brighton, who ran out 2-1 victors over Reading, while on Saturday Arsenal inflicted a 4-0 thumping on north London rivals Tottenham in front of a record WSL crowd.