Lewis Dunk rescued a late point for Brighton in his 400th league appearance for the club, scoring a towering header in the 95th minute.
Everton looked to be heading to an unlikely away victory before Pascal Gross crossed with his left foot to Dunk who climbed the highest and headed home past Jordan Pickford, who had no chance.
In his post-match press conference Roberto De Zerbi said: “Lewis is a legend, not just a captain, he deserves all the respect he gets from the club and the fans.
“I think he could play at a higher level, even in the Champions League.”
The draw means Brighton extend their unbeaten run at the Amex to 11 premier league games, a club record.
The 72nd minute saw the return of Julio Enciso from injury, but moments later Everton had broken the deadlock, with Pickford’s deep free-kick eventually ending up at the feet of Jarrad Branthwaite who took one touch before rifling home with his left foot into the top corner.
Brighton’s task of equalising was made even harder when Billy Gilmour was shown a straight red card in the 81st minute for a poor tackle on Amadou Onana, with the Everton man lucky to escape serious injury.
However, nine minutes of injury time allowed Brighton to throw more bodies forward and ultimately Everton were unable to hold on.
De Zerbi conceded that Brighton have been unlucky with injuries so far this season and that Gilmour will be another significant miss through suspension, saying:
“We have too many injuries, without [Kaoru] Mitoma, it is difficult for us. We are missing too many important players, which explains why we are drawing so many matches.
He added: “Billy [Gilmour] is a unique player, our only real senior playmaker in the squad.”
Substitute Beto could have put the game to bed for the visitors when he was played through in the 83rd minute, but he blasted wastefully over.
After the game, Sean Dyche said: “The second half was a great away performance, we controlled the game without the ball and kept Brighton out.
“I was worried when I saw the nine minutes injury time, you could feel the atmosphere in the stadium change and we didn’t deal with that as well as we could have.
“But Brighton have only lost one home game this season, and so I’m pleased to get a point.”
Everton have now failed to win any of their last nine Premier League games, the longest current winless sequence in the top-flight.
Brighton had several chances to take the lead in the first half, the best of them falling to Danny Welbeck after he was slipped in by Facundo Buonanotte, but he stabbed agonisingly wide past the on-rushing Pickford.
Everton continued to soak up the pressure in the second-half but Brighton had Tariq Lamptey to thank for keeping the score line goalless before he was substituted, heading the ball off the line from Abdoulaye Doucoure’s volley.
With the points shared Brighton remain in seventh and in the final European place. Everton moved one point ahead of Luton in 18th-place, having played one game more.
