I have successfully configured a VPN connection to my home router, thus, I was able to access machines in my private LAN. Moreover, I wanted to tunnel my whole Internet traffic over this connection. Even a google search could not answer, how to solve this with the network-manager which is used in most Linux desktops.
Finally, I figured it out. You need to go into the configuration of the particular VPN, further into the tab IPv4-Configuration, click on button routes and there you have to remove the tick from box which is probably named in English something like “Use this connection only for resources of current network”. German screenshots are attached below.