The Blaugrana could not build on their opening weekend’s victory over Bilbao’s local rivals Real Sociedad as they dug deep to secure a point in Basque.
Inigo Martinez brought the game to life on the 50th minute when he rose highest to head the hosts in front following a bright start to the second half.
Memphis Depay equalized for the visitors with his first goal for the club 15 minutes before the final whistle.
In an interview with the media after the game, Koeman believes his men made things difficult for themselves but admits it was a fair scoreline for both sides.
"It was a great game for many things in football. I think we made things too difficult for ourselves at the start of the game," said Koeman.
"They pressed us a lot and well. We need to know when we can play short, from the keeper as well. We weren't calm enough in the first 15 minutes.
"I think a draw is a fair result, we had chances to win, Athletic were at a very high level. Playing like that, this is a very tough ground, so I can't be unhappy with the result."
Koeman and Barcelona will entertain Getafe at the Camp Nou in their next La Liga game which will take place on August 29.
