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.
1. Robert Sanchez 6/10
Very nearly caught out by Richarlison after a heavy first touch from Dunk's pass back in the first half. Otherwise sound and a sharp second half save to deny Rodriguez at the near post
2. Ben White 7/10
Always looked in control of his game. Good defensive work and always clam and efficient in possession. Comfortable evening.
3. Lewis Dunk 7/10
The skipper was unlucky to see his instinctive header tipped over the bar in the second half. Defended well and dealt with Richarlison well
4. Joel Veltman 7/10
Still haven't seen Veltman have a bad game this season. Once again filled in on the left and looked comfortable as ever. Very good!