I know that Superman has been deputized by the Mayor of Metropolis (I believe it was in Byrne's 'Man of Steel' but feel free to correct that if I'm wrong) but Supes doesn't operate just in Metropolis, or even just in America for that matter. Is he, or the Justice League recognized as a law enforcement entity by the United Nations, thereby granting all it's members the ability to perform such tasks as arrest criminals? Technically he is a vigilante so any criminal he apprehends should be let go once official authorities arrive, unless he has some L.E. powers granted by the governing body of the realm right?
I understood most heroes to essentially be making citizen's arrests, which are permitted in most countries (to varying degrees and powers).
Even if we define heroes as vigilantes, there is no need for criminals/villains to be cut loose just because they weren't caught by law enforcement. The proper law enforcement officials are still required to make the actual, official arrest and perform all the actions that that entails (reading of rights, etc).
In addition to the posited Citizen's Arrest, and the ability for deputized persons to make arrests out of their local jurisdiction, there is the concept of Hot Pursuit. The WP article only mentions it in a nautical sense, but I'm sure I've heard of its use allowing law officers to cross state lines in the US.
Ultimately, though, the answer may well just come down to the fact that he is Superman, and that the DC universe where Metropolis and Superman exist has slightly different laws to our world.
While mainstream comic book universes often feign attempts to indicate that they take place within a reality that is analogous to our own, in the realm of American Criminal Justice, there's simply no way to reconcile the two across the board without making significant alterations to American jurisprudence as we know it.
In the criminal justice system, the burden of proof lays with the State to establish "beyond all reasonable doubt" that the accused is guilty of the alleged offense. That standard of proof would be nearly impossible to achieve in the Marvel and DC Universes where there is documented proof of the supernatural, shape-shifters, reality warpers, mind control, and a whole litany of physics-defying powers, abilities, and realities. The triumvirate of prosecutorial proofs- means, motive, opportunity- are made a mockery of by the powers, tech, and magic afforded perpetrators in mainstream comics.
Therefore, in order to maintain any semblance of law and order as we know it, there MUST be laws on the books of the DCU to address the paranormal, alien, mystical, and more. Likely, there is some burden shifting to the defense to assert the paranormal as an affirmative defense, but the bottom line is that their reality would have adapted to the issue by enforcement, case law, legislation, etc. That is what law is meant to do, bring order by way of self-modification and discretion to address a given situation.
Applied to Superman, et al.
It is an untenable situation to- in the name of due process- allow supervillains and ordinary criminals run free simply because they were apprehended by a superhero rather than an ordinary one... and, in the case of the supervillain, improbable that capture is even possible without the assistance of the superhero. Thus it is incumbent on society to tolerate or regulate superhero intervention, even at a minor loss of due process, in order that society go on existing and not suffer at the hands of supervillains! So most likely there are provision, ordinances, laws, treaties, or executive orders providing the relevant police powers or jurisdiction to Superman where invited.
In short, it only seems unreasonable if you assume American law wouldn't change, but in light of what superheroes do to the justice system by their mere existence, I absolutely believe law the globe over would have to change to accommodate their reality... making it fine for Superman to arrest beyond his typical jurisdiction.
Most states have laws that a properly deputized sheriff's deputy can make arrests throughout the state when they see felonies committed in plain view.
On top of that, you have to consider that these heroes were invented at a time in US culture where we were considered to be the dominant superpower, and essentially the world police. A lot of that shows in their behaviors and personalities.