Saints looked as if they were heading for a third successive defeat in the Barclays Premier League when José Fonte was sent off in the 79th minute and Jermain Defoe then put the Black Cats ahead five minutes from time.
But the hosts responded and earned themselves a point deep into added time when Virgil van Dijk struck home an equaliser.
Koeman explained that the attitude shown by his players in those final minutes had satisfied him.
“We kept going and we showed great character after going 1-0 down, ten against 11.
“Luckily we scored the 1-1 and it was a great goal and a great attack and a great finish from Virgil.”
Koeman added: “That shows that we still believe and have belief to win the game, or even with one less on the pitch to have one point out of this game and that's good, but we need to come back in the team, in the confidence and to show from the first minute.
“Today was really struggling, really difficult, because too many players far away from their best level.”