Although seeing that they are both Supernovas, and therefore rivals, there may be some ulterior motive behind Law’s decision to save Luffy. While on the other hand, Law has been shown to respect Luffy, and could quite simply have come to save him because of that respect. When Luffy met Law on Punk Hazard, Luffy was happy to see him again, not knowing Law’s intentions. Luffy stated, with a happy smile on his face, that he and Law will be enemies if he’s going after the One Piece too, but he is grateful nonetheless.
When Law helped save Nami, he proposed an alliance to Luffy to take down one of the Yonko. To the surprise of their crewmates, he accepted after being told that they would be taking down Kaido saying he intended to take them all down.
Luffy appears to trust Law to go along with this plan of action. Due to Law being his life saviour, Luffy has great gratitude towards Law, considering Law to be his friend. While Luffy perceives this alliance as practically a friendship, Law sees it more as a professional agreement.
Since the two have formed an alliance, Luffy has developed a running gag of making decisions on behalf of the alliance without considering Law’s opinion, usually to the latter’s annoyance. He decided to send Caesar Clown flying away in Punk Hazard when the agreed upon plan was capturing him, defeating Donquixote Doflamingo outright when the plan was leaving him for Kaido. They even have a comical argument about this in Dressrosa, as Law decides to terminate the alliance in light of their plan being compromised. Luffy says Law is his friend and that he can’t decide these kinds of things for the both of them, and Law retorts that is precisely what Luffy is doing. When Luffy agrees to help the Mink tribe and the Kozuki clan take down Kaido, Law feels slighted at not being asked for input, questioning Luffy about his faith in him.