Eliezer Mayenda gave Sunderland a first-half lead at Dean Court before Evanilson struck an equaliser on 63 minutes.
Speaking post-match, Le Bris said: “Bournemouth are intense - it has been developed over many years.
“They are duel-driven and always on the front foot. It was really positive to challenge this situation after three defeats in a row. It is not a question of tactics.
“It is a question of mindset, and today we showed we had the opportunity to win this game, especially at the beginning of the second half.
“We didn’t score, but it is a good point in the end. We have different options on the bench and we picked up a few injuries against Fulham, so it was important to change the squad.
“We have a small group, but they are all connected and they know that with injuries and suspensions, opportunities will come.”
He continued: “(Eliezer) is in the right location on the pitch, so as a striker you have to sniff the opportunity, and he did it well today.
“(Granit Xhaka) is important because in this league you need to have two or three main players, and Granit is one of those for us.
“You can feel the difference when he is here with us. He has quality, first of all, but he can give his team-mates confidence and drive them forward.”