Secondly, from you question, I am not sure if you want to move files from:
a SharePoint-hosted app's web to the host web => If you want to move files from the app web to the host web using JS, there is no other way than using the SP.RequestExecutor object.
or from a standard SharePoint web to another standard web => In this case you can simply use AJAX (jQuery, SuperAgent, etc.) and the RESTful URL below. Be aware this works only when both webs are on the same domain. If you want to move files between two webs on different domains, try using a SharePoint 2013 workflow and the call web service action.
Here is the Uri structure for the SP.File.moveTo method:
http://<sitecollection>/<site>/_api/web/folders/getbyurl(folderrelativeurl)/files/getbyurl(filerelativeurl)/moveTo(newUrl,flags) as taken from here.