Everton full-back Leighton Baines pulled one back late on with a penalty as Palace were forced to hold on for all three points.
Below, Sports Mole takes a closer look at the key talking points from the match.
On target: 6
On target: 3
Was the result fair?
On the balance of play, Everton really should have taken something from this match. They enjoyed long spells on top as the statistics suggest, but a lack of focus at the back ruined their chances of avoiding defeat on a dramatic afternoon at Goodison Park.
Everton were very good in spells, but individual mistakes cost them dearly. As expected, they dominated possession and always looked comfortable on the ball. They were perhaps guilty of lacking a cutting edge in the final third as they suffered their first Europa League hangover of the season. Tim Howard made two errors which led to goals, while a number of other Everton defenders, including John Stones, produced below-par displays.
Crystal Palace's performance
Neil Warnock will be absolutely delighted with how his side battled against a team lauded for their ability to keep the ball for long spells. Palace recovered from a sluggish start, and grew in confidence after Mile Jedinak powered a penalty home. They spent much of the match chasing the ball, but refused to gift the hosts time to make decisions in and around the box. They defended admirably, and were rewarded for being positive on the break.
Sports Mole's man of the match
Yannick Bolasie: On another day this award may have gone to Scott Dann, who performed superbly at the back before giving away a late penalty. Bolasie was superb throughout as he troubled the Everton defence with his pace and trickery. Rounded off an impressive display with a good goal, too.
Stones and Howard both walk away with this dubious honour. A ball into the box should have been easily cleared in the first half, but the Everton pair hesitated, and Howard eventually gave away the penalty with a foul on McArthur.
Michael Oliver did very well this afternoon. He was always willing to let the game flow, and made the correct decision with both penalties.
Everton: The Toffees return to action against Swansea City in the Capital One Cup on Tuesday night.
Crystal Palace: Warnock's side play 24 hours later against Newcastle United in the same competition.