It's quite simple really: The US cannot prevent losing control, but they can have it happen in an orderly way and perhaps get a better position in the resulting system.
You see, it's not like there is some magical Key To the Internet which is stored in a bunker in Oregon and which you can choose to either hand over or not. It's also not something you really can defend with guns to prevent other countries from having it.
It's rather more like having control over the rules of international air traffic. If you do it well and neutrally enough, it might be that few countries are annoyed that they don't have a say in the process you have set up for writing the rules. But you have no way of really enforcing those rules except inside your own borders.
Currently ICANN which drafts the rules (and works as the judges) for the Internet is for historical reasons set up as a US entity. It having control over the Internet means no more and no less than all countries deciding to implement their decisions.
The reason why ICANN still has control and the reason for this statement by the EU is that other countries are still hoping for a negotiated solution, because that's generally the way the civilized world works. The US might be in a slightly better position to negotiate than other countries, but if it refuses to negotiate, it will surely lose that advantage. An orderly solution would be in everybody's interests, while more unilateral action would harm everyone.
The orderly way to proceed would be to continue with ICANN, just internationalized. The disorderly way might be setting up a parallel organization and start disregarding ICANN.
Still you must realize it's a pipe dream that a single country with a few percent of world population could keep the right to make the rules for much longer. So sad you Americans feel offended about this. The rest of the world doesn't really think it's even asking for anything that in any meaningful sense belongs to you when they ask to have a say.