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.
5. Pascal Gross 6/10
Corners and set pieces were a constant danger to Everton's defence. Didn't have so impact in open play as most Brighton attacks seemed to arrive from the left
6. Yves Bissouma 8/10
Impressed throughout the 90 an an accomplished display. Unlucky with a spectacular attempt from an overhead kick in the first first. Drew priase from Gary Lineker who tweeted: "What an exceptional footballer is @Yves_Bissouma"
7. Adam Lallana 7/10
A polished display in the centre of the park and adds control to Brighton's midfield. Pressed well in first half and good in possession. One efforts in the second half whistled just over
8. Jakub Moder 8/10
The Poland international has slotted seamlessly into this Albion side. Played down the left and narrowly off target with a two decent first time efforts