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.
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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.