David Garcia and Chimy Avila both scored for Osasuna.
"I think that's the positive and the negative," Barca coach Xavi Hernandez said after the game of his young goalscorers.
"The players making the difference at Barcelona now are 17, 18 and 19. Positive for the future but also a negative. Such young players won't always perform excellently. We need time, patience. Gavi, Nico were spectacular. But they don't want to be the spine of the team."
Gerard Pique said that the club "urgently" needed a win to get their season started.
"Like Xavi said [after the defeat to Bayern on Wednesday], a new era is starting and it's vital we start picking up points in LaLiga," Pique told Movistar.
"We're adrift [of the top four] and the more points we can make up in the coming weeks, the better. We competed today, but the result was a huge shame.
There was an early flurry of goals with Gonzalez scoring after just 12 minutes. The 19-year-old was one-on-one with goalkeeper Sergio Herrera and calmly slotted the ball into the net.
Barca's lead didn't last long, though, as it took Osasuna just two minutes to score an equaliser. The home side won a free-kick and Garcia headed it into the net.
The half continued quietly without many chances for either side but burst into life again four minutes into the second half when Ezzalzouli put Barca ahead.
Xavi's side caught Osasuna on the counter-attack and while Garcia got a foot to the ball in the box, he only flicked it as far as Ezzalzouli who smashed it into the roof of the net.
It looked like Barca had done enough to secure the win but Avila scored a late goal in the 85th minute to give Osasuna a draw. The striker secured the ball from a corner and powered into the net.
Gonzalez's and Ezzalzouli's goals mark the first time this century that Barca have had two different players under the age of 20 score in the same LaLiga season.
Xavi said the club needed "a reset," to create a new Barca.
"We are in a negative spiral and we need to find a way to break it," he added. "This draw is not enough. When we scored the second goal we should have taken control but we didn't and that made us suffer."