The Eagles and the Toffees shared the spoils in their opening game as it finished a barren draw.
Both sides had little to show in the form of goalscoring chances in the first half, but produced more attacking threat after the break.
Jordan Pickford denied Jordan Ayew and Max Meyer in quick succession, and on the other end, Guaita stopped a long range effort from Gylfi Sigurdsson.
With 15 minutes to full time, Morgan Schneiderlin caught Luka Milovojevic with a late challenge from behind, and the Everton man got his second booking and ultimately his marching orders.
Everton were then happy to keep possession and mostly passed the ball between their defenders to run the clock down. Palace had a few chances in between but could not produce a decent final touch.
Wilfried Zaha, who is unhappy not to have sealed a move away from Palace this summer, came on in 67th minute, while Moise Kean made his Everton debut after replacing Dominic Calvert-Lewin in the 71st minute.
Marco Silva would probably be happy with the clean sheet, but will be hoping his men can show more in an attacking sense when they host Watford on Saturday 17. Meanwhile, Palace travel to Sheffield United.