By default ubuntu 18.04 is using systemd-resolve. In order to use openvpn, resolvconf and dnsmasq you need to do following:

Ensure that all necessary packages are installed:

apt-get install -y openvpn resolvconf dnsmasq

NetworkManager will not control dnsmasq so we need to add following config:

echo -e "[main]\ndns=none" > /etc/NetworkManager/conf.d/dnsmasq.conf

Disable and stop systemd-resolved:

systemctl disable systemd-resolved.service
systemctl stop systemd-resolved.service

Enable and start dnsmasq:

systemctl enable dnsmasq.service
systemctl start dnsmasq.service

Network settings including nameservers to use are managed by the NetworkManager. In order for NetworkManager to notify resolvconf that settings need to be updated /etc/NetworkManager/dispatcher.d/resolvconf should be added:

#!/bin/bash

case "$2" in
  up)
    [ -z "$IP4_NAMESERVERS" ] && exit 0
    for x in $IP4_NAMESERVERS; do
        echo "nameserver $x" | /sbin/resolvconf -a "$1.networkmanager"
    done
    ;;
  down)
        /sbin/resolvconf -d "$1.networkmanager"
    ;;
  *) exit 0 ;;
esac

Modify /etc/resolvconf/update.d/dnsmasq (please note “openvpn support” comments):

#!/bin/sh
#
# Script to update the resolver list for dnsmasq
#
# N.B. Resolvconf may run us even if dnsmasq is not (yet) running.
# If dnsmasq is installed then we go ahead and update the resolver list
# in case dnsmasq is started later.
#
# Assumption: On entry, PWD contains the resolv.conf-type files.
#
# This file is part of the dnsmasq package.
#

set -e

RUN_DIR="/run/dnsmasq"
RSLVRLIST_FILE="${RUN_DIR}/resolv.conf"
TMP_FILE="${RSLVRLIST_FILE}_new.$$"
MY_NAME_FOR_RESOLVCONF="dnsmasq"

[ -x /usr/sbin/dnsmasq ] || exit 0
[ -x /lib/resolvconf/list-records ] || exit 1

PATH=/bin:/sbin

report_err() { echo "$0: Error: $*" >&2 ; }

# Stores arguments (minus duplicates) in RSLT, separated by spaces
# Doesn't work properly if an argument itself contains whitespace
uniquify()
{
	RSLT=""
	while [ "$1" ] ; do
		for E in $RSLT ; do
			[ "$1" = "$E" ] && { shift ; continue 2 ; }
		done
		RSLT="${RSLT:+$RSLT }$1"
		shift
	done
}

if [ ! -d "$RUN_DIR" ] && ! mkdir --parents --mode=0755 "$RUN_DIR" ; then
	report_err "Failed trying to create directory $RUN_DIR"
	exit 1
fi

RSLVCNFFILES=""
for F in $(/lib/resolvconf/list-records --after "lo.$MY_NAME_FOR_RESOLVCONF") ; do
	case "$F" in
	    "lo.$MY_NAME_FOR_RESOLVCONF")
		# Omit own record	 
		;;
	    lo.*)
		# Include no more records after one for a local nameserver
		RSLVCNFFILES="${RSLVCNFFILES:+$RSLVCNFFILES }$F"
		break
		;;
          # start of the openvpn support change
	  *.openvpn)
		RSLVCNFFILES="$F"
		break
		;;
          # end of the openvpn support change
	  *)
		RSLVCNFFILES="${RSLVCNFFILES:+$RSLVCNFFILES }$F"
		;;
	esac
done

NMSRVRS=""
if [ "$RSLVCNFFILES" ] ; then
	uniquify $(sed -n -e 's/^[[:space:]]*nameserver[[:space:]]\+//p' $RSLVCNFFILES)
	NMSRVRS="$RSLT"
fi

# Dnsmasq uses the mtime of $RSLVRLIST_FILE, with a resolution of one second,
# to detect changes in the file. This means that if a resolvconf update occurs
# within one second of the previous one then dnsmasq may fail to notice the
# more recent change. To work around this problem we sleep one second here
# if necessary in order to ensure that the new mtime is different.
if [ -f "$RSLVRLIST_FILE" ] && [ "$(ls -go --time-style='+%s' "$RSLVRLIST_FILE" | { read p h s t n ; echo "$t" ; })" = "$(date +%s)" ] ; then
	sleep 1
fi

clean_up() { rm -f "$TMP_FILE" ; }
trap clean_up EXIT
: >| "$TMP_FILE"
for N in $NMSRVRS ; do echo "nameserver $N" >> "$TMP_FILE" ; done
mv -f "$TMP_FILE" "$RSLVRLIST_FILE"

Enable resolvconf:

resolvconf -u

Restart NetworkManager:

service NetworkManager restart