Everton began the day only seven points in front of Palace and in danger of being dragged into a relegation battle.
But the victory took Brian Sorensen's side 10 clear and up a place, above Tottenham.
Palace remain in trouble. They won promotion from the Championship last season, but sacked manager Laura Kaminski at the end of February, with Leif Smerud replacing her.
The Norwegian lost his first two games in charge, before a vital 3-1 win over Villa for only Palace's second league win of the season.
Smerud's side had opportunities to take the lead at Walton Hall Park with Canadian Clarissa Larisey a threat down Palace's left wing.
One cross found Ashleigh Weerden but her shot on the spin was saved by home goalkeeper Courtney Brosnan, while another dangerous pass into the penalty area was superbly defended by Vanhaevermaet.
Everton had a great deal of possession but their 43rd-minute goal, which came from their first notable effort, extended their run of scoring in home WSL matches to six successive games.
Palace instantly fell further behind at the start of the second half, with Holmgaard adding a goal to her earlier assist.
There was controversy shortly after as Vanhaevermaet looked to have clipped the heels of American forward Stengel. The referee initially pointed to the spot but then allowed Stengel, who had stayed on her feet, the advantage, but the chance then came to nothing.
Everton were able to bring on Denmark forward Rikke Madsen as a late substitute for her first appearance of the season after giving birth to a daughter in the autumn, before Gago's third goal late on.
It proved to be a memorable afternoon for Madsen. At the end of the game, her boyfriend Martin Thomsen went down on one knee on the pitch and proposed, with the 27-year-old accepting.
Palace's relegation rivals, Villa and Leicester, are at home on Sunday, against Manchester United and Brighton respectively.
The Eagles are running out of games to save themselves and have matches against the current top two sides next. They entertain second-placed Arsenal on 30 March before a trip to unbeaten leaders Chelsea on Sunday, 20 April.