Iliman NDiaye gave the visitors an early lead after just 15 minutes with a brilliant solo goal. But Sunderland's captain Granit Xhaka inspired the comeback, levelling the match just 45 seconds into the second half with a deflected strike from the edge of the box.
The result extended Sunderland's unbeaten home run in the Premier League this season and moved them into the top four, having now collected a league-high eight points from losing positions.
Everton, meanwhile, have won just one of their last eight matches in all competitions.
Le Bris said he was proud of the way his players responded after a slow start. "It is a good point in the end," he told the club's website. "We didn't start well, and they were better than us for the first 30 to 35 minutes. They have many threats and were dangerous when we lost the ball, but we improved after the break.
"It was really important to score early in the second half. That goal gave us energy and confidence. Granit was exceptional, he drove us forward and showed his quality again.
"The connection between the team and the fans was special, and it pushed us to fight for every ball."
Everton had their chances to extend their advantage before Xhaka's equaliser, with Jack Grealish striking the post and Thierno Barry missing from close range. But once Sunderland found their rhythm, they controlled large spells of the match, with Enzo Le Fee and Dan Neil dictating play in midfield.
Although Sunderland could not find a winner, Le Bris said he saw enough to believe his side can continue to compete at the top end of the table. "We will face different moments this season," he added.
"What matters is how we react. Tonight, we showed character, and that is very important."
Sunderland will look to build on their strong form when they host Arsenal on Saturday, while David Moyes' Everton will aim to bounce back at home to Fulham on Saturday.
Mikel Arteta has admitted Arsenal must remain grounded despite their stunning start to the season, as the Gunners continue to battle on multiple fronts at home and in Europe.
Tottenham manager Thomas Frank has eased fears over injuries to Mohammed Kudus and Djed Spence ahead of Tuesday's crucial Champions League clash with Copenhagen.
Liverpool manager Arne Slot has backed Florian Wirtz to come good despite the German midfielder's difficult start to life at Anfield.
Everton manager David Moyes admitted his frustration after his side failed to turn a dominant start into victory in their 1-1 draw against Sunderland at the Stadium of Light.
