Brighton always looked the team more likely to sneak the winner but they just couldn't find a way through a stubborn Everton rearguard.
Neal Maupay, Danny Welbeck, Adam Lallana, Yves Bissouma and Jakub Moder all went close for Graham Potter's team who continue to edge their way to Premier League safety.
Everton were without their main striker Dominic Calvert-Lewin and struggled to create chances but they came so close to winning the match at the end when Alex Iwobi cut in from the right and flashed an effort just over the bar.
Scroll down and click through to see how Brighton players rated.
9. Leo Trossard 6/10
A quieter performance from the Belgian. Tried to influence the attack after the break but just couldn't find a way through the Everton defence
10. Neal Maupay 6/10
Once again worked tirelessly in a more central role but missed the best chance of the match when he snatched his right footed effort wide in the second half
11. Danny Welbeck 6/10
Attacked off the left and looked a threat but just couldn't find a way through. Had one effort in the second half excellently blocked by Everton's Ben Godfrey. Frustrating evening!
12. Ali Jahanbakhsh 6/10
On for Maupay but no chance to make an impact