No, America's way is more using direct interventions or to fund rebel groups to destabilise the region.
Russia does that too, though, so I wouldn't really call it specifically America's way. And they also annex the land. Yeah, America does some of the things that Russia does, but it doesn't do anywhere close to all the shit that Russia does. On the other hand, if America does a bad thing, you can be sure Russia has done it and then some.