So, this is basically a foundational plot point that's seeded early but confirmed quite late, and honestly, it's done through a pretty satisfying chain of reveals. The first major piece of evidence is Naruto's name itself. 'Naruto' comes from the name of the hero in Jiraiya's first book, which is a story deeply personal to Minato. Minato essentially named his son after the main character in his mentor's novel, which is a piece of trivia only someone close to Jiraiya would know. The big info-dump confirmation comes from Jiraiya himself during the 'Search for Tsunade' arc, where he tells Naruto about his parents. He reveals Minato sealed the Nine-Tails into Naruto to make him the jinchuriki, and that was the last act of the Fourth Hokage, sacrificing himself for the village.
Further proof comes from the Third Hokage's interactions. He's shown to have a special, almost grandfatherly concern for Naruto, which makes sense if he's watching over his predecessor's orphaned son. The physical resemblance is also a clue—the spiky blond hair is unmistakable. But the absolute, no-doubt confirmation is the visual during the Pain arc when Naruto loses control. We see a vision of Minato and Kushina within the seal, and Minato explicitly calls Naruto 'my son' and explains his actions. That scene removes any lingering ambiguity.