For those of you that are interested, I managed to solve this myself.
Apparently, "ipt_MASQUERADE," the module that makes masquerading possible, is not (yet?) available in OpenVZ. The absence of this module is what causes the following command to fail:
root@xx:~# iptables -t nat -A POSTROUTING -d x.x.x.x -j MASQUERADE
iptables: No chain/target/match by that name.
I discovered that it was still possible to accomplish my goal, only that an alternate second command was required. So here's what I did.
1. Cleared out existing nat PREROUTING and POSTROUTING rules from earlier testing.
root@xx:~# iptables -t nat -F PREROUTING
root@xx:~# iptables -t nat -F POSTROUTING
2. Added the following two rules:
root@xx:~# iptables -t nat -A PREROUTING -p tcp -d x.x.x.x --dport 7777 -j DNAT --to y.y.y.y:7777
root@xx:~# iptables -t nat -A POSTROUTING -j SNAT --to-source x.x.x.x
*Where x.x.x.x represents the source IP and y.y.y.y represents the destination IP.
3. Saved iptables (necessary for changes to persist after reboot)
Hope this helps someone...I know it would have helped me.