TL;DR The Third Hokage, Sarutobi Hiruzen, looked after Naruto, in accordance with the Fourth Hokage's wish to have the village see Naruto as a hero.
Early in the series, Sarutobi mentions how the Fourth Hokage sealed the Kyuubi inside Naruto, and died wishing that the village would treat Naruto as a hero. Much later, it is also revealed that:
Sarutobi personally witnessed the Fourth Hokage, Namikaze Minato, use the Shiki Fuijin technique to assist in sealing the Kyuubi inside Naruto. Sarutobi remarks about Minato making his own son a Jinchuriki to save the village. In the process, Naruto also became an orphan minutes after he was born. (Chapter 504)
Sarutobi saw the Fourth Hokage's deed as a heroic sacrifice (especially in the light of the spoiler-text above), and took up the responsibility of fulfilling the Fourth Hokage's wish. Naturally, that includes paying for the orphan child's expenses until he is old enough.
Along the same lines, it was also mentioned that Sarutobi banned any discussion of the Kyuubi incident and severely punished anyone who disobeyed. Most other people either hated Naruto or were indifferent towards him, as he remarks multiple times later on during the series. Thus, nobody besides Sarutobi could possibly care to pay for Naruto's expenses.1
Hence, although nobody explicitly stated that the Third Hokage looked after Naruto's expenses (room rent, food, clothes, etc.), it is obvious that this is what happened.
1 This also leads to one of the many plotholes in Naruto's childhood story: Naruto has said a few times that Iruka-sensei was the first person who cared for him, and before then, his life was a hell of loneliness. Why didn't Sarutobi treat him with a bit more warmth from the beginning, especially when he knew who Naruto's parents were?