Ubuntu – OpenVPN for certain IPs, eth0 for everything else

Summary: I'd like to connect to my VPN and have access to certain servers, but for all other traffic I'd like to use my regular networking.

I've setup an OpenVPN server on my VPS, my server.conf file looks like so:

port 1194
proto udp
dev tun
ca ca.crt
cert server.crt
key server.key  # This file should be kept secret
dh dh2048.pem
server 10.8.0.0 255.255.255.0
ifconfig-pool-persist ipp.txt
keepalive 10 120
comp-lzo
user nobody
group nogroup
persist-key
persist-tun
status openvpn-status.log
log         /var/log/openvpn.log
verb 4
push "route 10.132.0.0 255.255.0.0"

I use the following .ovpn file to setup the VPN Connection:

client
dev tun
proto udp
remote <my.vpn.server.com> 1194
nobind
user nobody
group nogroup
persist-key
persist-tun
remote-cert-tls server
comp-lzo
verb 3
<ca>....</ca>
<cert>...</cert>
<key>...</key>

Finally, in the Network Manager for the VPN Connection, under IPv4 Settings I have made sure to set the "Method" to "Automatic (VPN) addresses only".

VPN connects fine, I can access all the internal servers I need (10.132.x.x), however I cannot access anything else (like google.com). I'd like my eth0 settings to be used for everything except for the 10.132.x.x IPs which I would like routed through the VPN.

P.S. based on other articles I've tried using no-pull in the .ovpn file and adding in my route settings there but to no avail.

EDIT 1:

Results of running ip a and traceroute while connected to VPN:

$ ip a
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1
    link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
    inet 127.0.0.1/8 scope host lo
       valid_lft forever preferred_lft forever
    inet6 ::1/128 scope host 
       valid_lft forever preferred_lft forever
2: eth0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc pfifo_fast state UP group default qlen 1000
    link/ether 08:00:27:dc:a6:ef brd ff:ff:ff:ff:ff:ff
    inet 10.0.2.15/24 brd 10.0.2.255 scope global dynamic eth0
       valid_lft 86320sec preferred_lft 86320sec
    inet6 fe80::f3d1:6eb3:e13e:d61b/64 scope link 
       valid_lft forever preferred_lft forever
15: tun0: <POINTOPOINT,MULTICAST,NOARP,UP,LOWER_UP> mtu 1500 qdisc pfifo_fast state UNKNOWN group default qlen 100
    link/none 
    inet 10.8.0.6 peer 10.8.0.5/32 brd 10.8.0.6 scope global tun0
       valid_lft forever preferred_lft forever

$ traceroute google.com
google.com: Temporary failure in name resolution
Cannot handle "host" cmdline arg `google.com' on position 1 (argc 1)

EDIT 2: results of ip r

$ ip r
default via 10.8.0.5 dev tun0  proto static  metric 50 
default via 10.0.2.2 dev eth0  proto static  metric 100 
10.0.2.0/24 dev eth0  proto kernel  scope link  src 10.0.2.15  metric 100 
10.8.0.1 via 10.8.0.5 dev tun0  proto static  metric 50 
10.8.0.5 dev tun0  proto kernel  scope link  src 10.8.0.6  metric 50 
10.132.0.0/16 via 10.8.0.5 dev tun0  proto static  metric 50 
104.236.239.153 via 10.0.2.2 dev eth0  proto static  metric 100 
169.254.0.0/16 dev eth0  scope link  metric 1000