Justice isn't something that's set in stone in Harry Potter and the Prisoner of Azkaban. It isn't the hardcore justice we see in Justified, the final "dun-dun" judgments we get in Law and Order, or even the vigilante justice of something like Batman in The Dark Knight. In fact, in the Harry Potter series, justice and punishment are issues that are up for debate. The scene that best demonstrates this is when Harry asks his father's friends to spare Peter. There are competing ideas of justice swirling around here – we have justice as vengeance, justice as mercy, justice as punishment, and justice as forgiveness (phew!).
We can trace this theme backwards throughout the entire novel too – Sirius's imprisonment and punishment, Buckbeak's trial, Ron's grudge-match punishment of Hermione. Justice is hard to mete out but it's a responsibility that Harry and other characters have to struggle with throughout Book 3. As Gandalf tells Frodo in The Lord of the Rings, you shouldn't be too eager to cast judgment or dole out punishment for others. This is a lesson that a furious Harry, longing for revenge for the huge injustice of his parent's murder, has to learn over the course of the novel.