Not sure what you mean by allowing apps to access internet. Usually its the other way round where you want to restrict certain apps. A lot of apps are now available that control connection of apps. For example restrict to wifi only. Even Google Play Store have that option where apps are updates only on wifi. Onavo has options to control internet usage of apps.
If you off your data connection, then no internet. I don't you would want an app to on the data connection and connect to the net on its own. SO usually, you leave the data connection on, and find an app to restrict data for certain apps.
