If your OpenVPN server pushes DNS nameservers and search domains to the client connections, your OpenVPN client should be installing and using them.

If this does not happen, review your OpenVPN client security settings. You might be constrained to not allow running external scripts and you will be able to spot the error straight away.

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When the client receives connection parameters, it parses DNS being sent by the server. Once it has it all, the client will try to run the script at /etc/openvpn/update-resolv-conf that will add the received config on top of your current /etc/resolv.conf file. External script run security constraint will prevent this from happening.

Easy solution in this case. Just add the following to the OpenVPN config file:

--script-security 2