It sounds like you will want to use the Spatial Adjustment tools in ArcMap to accomplish this.
Take a look at the following documentation link to learn a bit about the tool:
The Spatial Adjustment toolbar will let you essentially conduct vector georeferencing using the control points data sets you have. You will need to specify spatial adjustment transformations. See here for documentation explaining the differences between the methods:
From what I understand of your situation you will likely want to use a Similarity transformation method which scales all features as one (rather than independently). Additionally it only requires you to have a minimum of two control points. The tool does recommend that you have three or more control points though to increase accuracy.
In order to use the tools, it should be as simple as opening both the Editor and Spatial Adjustment toolbars in ArcMap and entering an edit session for the target feature classes. Then you can use the Spatial Adjustment toolbar to test out some transformations.