Ilaix, 18, scored Barca's second goal with seven minutes to go to seal victory as Ronald Koeman's side cut the gap on La Liga leaders Atletico Madrid to just two points.
The midfielder, who joined the club's academy in 2010, only made his professional debut in January and was making just his fifth appearance for the senior side.
"I like to shoot," Ilaix told reporters after the game. "Thanks to God, [Lionel Messi's] passed me the ball and I don't know how I cut back, but then I shoot with my left [foot] and it's a goal.
"I need a little time to take it all in. I will never forget it. I will take it to the grave with me. I just want to dedicate the goal to my dad, to my mum... and to all my family."
He came through the La Masia academy at the same time as Ansu Fati, whose first Barca goal also came away against Osasuna last season.
"Ansu is like my brother, we've spent many years together," Ilaix said. "I love him a lot, he's family to me. I'm delighted to have scored my first goal in the same goal as him."
Asked what advice Messi, who set up the goal, has given him, Ilaix added: "He's said to me from the start: 'You always have to the area. Get there and look for the goal.' And today he gave me the pass and I've shot..."
Ilaix has long been highly rated within the club -- he signed an unprecedented contract for an academy player in 2019 when he was just 16 -- but Koeman still deserves credit for the faith he's shown in him.
"Ilaix scored a fantastic goal, the decisive goal that sentenced the game," the Barca coach said in a news conference.
"Every time he comes on, he gives the team something extra, because of his personality and his energy. Most of all, he's good with the ball and physically he's a strong kid. He's gaining ground and showing he can be with the first team more.
"For me, the most important thing is that we're competitive. Age doesn't matter, it's about what I see on the pitch."
Koeman conceded his team weren't at their best but still felt they "deserved" to leave Pamplona with three points.
The Dutch coach will be an interested viewer on Sunday when leaders Atletico host third-placed Real Madrid in the Madrid derby.
Atletico's lead over Barca is down to just two points, although they have two games in hand, while Madrid are three points back from Barca but have played a game less.
"The most important thing was to win our game," Koeman said. "Whatever the result is on Sunday will be a good one for us."