Tell docker to use the DNS server in the host system

You can use the host's local DNS resolver (e.g. dnsmasq) from your Docker containers if they are on a user defined network. In that case a container's /etc/resolv.conf will have the nameserver 127.0.0.11 (a.k.a. the Docker's embedded DNS server), which can forward DNS requests to the host's loopback address properly.

$ cat /etc/resolv.conf
nameserver 127.0.0.1
$ docker run --rm alpine cat /etc/resolv.conf
nameserver 8.8.8.8
nameserver 8.8.4.4
$ docker network create demo
557079c79ddf6be7d6def935fa0c1c3c8290a0db4649c4679b84f6363e3dd9a0
$ docker run --rm --net demo alpine cat /etc/resolv.conf
nameserver 127.0.0.11
options ndots:0    

If you use docker-compose, it will set up a custom network for your services automatically (with a file format v2+). Note, however, that while docker-compose runs containers in a user-defined network, it still builds them in the default network. To use a custom network for builds you can specify the network parameter in the build configuration (requires file format v3.4+).