Depay found the net with a fine left-foot half-volley 15 minutes from time in the Basque Country for his first goal in La Liga since moving to Barcelona from Lyon.
Ronald Koeman's side had impressed in their first match of the post-Lionel Messi era as they beat Real Sociedad 4-2 last weekend, but they were fortunate to depart with a point from this game and it was impossible to avoid questions about the Argentine at full-time.
"I don't like to always talk about the same thing but he is the best in the world and the opposition might be more afraid if Messi is there," Koeman admitted.
"For us too, if there is a pass to Leo, normally he doesn't lose the ball, and today we gave the ball away too often in the first half.
"You notice it, but that's how it is, it can't be changed now."
It was a difficult night for Barcelona's defence in particular, with Gerard Pique coming off after half an hour and Eric Garcia being sent off in stoppage time for bringing down Inaki Williams.
Former Manchester City defender Garcia received news of his grandfather's death before the match and struggled throughout the 90 minutes, allowing Inigo Martinez to get away from him to head in the opener early in the second half.
"He is a great professional to play so well in a game after the death of his grandfather. We wish him all the best in his personal situation," said Koeman, who insisted the draw was the right result.
Athletic, though, will feel they should have won despite missing several key players, among them goalkeeper Unai Simon, who has not featured at the start of this season after helping Spain reach the Euro 2020 semi-finals and then the final of the Tokyo Olympics.
Barcelona were unchanged from their opening victory meaning another start for Pedri, the brilliant young midfielder who also played at Euro 2020 and the Olympics and is now set to get a two-week holiday.
Martin Braithwaite, who netted twice against Real Sociedad, missed an early sitter for the visitors when he blazed over after great play by Depay.
Athletic, with 11,000 of their fans allowed into the 53,000-capacity San Mames for this game as coronavirus restrictions are gradually loosened in la Liga, then saw Oihan Sancet turn and crash a shot off the crossbar from 12 yards out.