Taufua's 62nd-minute effort reduced the gap to 14 with just under 20 minutes remaining and, while tempers flared a number of times in the closing stages, the Storm held on to claim their 18th win of the year.
Melbourne prop Felise Kaufusi sealed victory with a try on fulltime, set up by Cooper Cronk's 25th try assist of the year, the most of any player in the NRL.
Vunivalu became the first player to bag four in one half since former Sea Eagles winger David Williams did it in 2012, and just the sixth Storm player to achieve the feat.
His first came when Cameron Munster darted a pass in the 10th minute, and bagged his second when he chased through a Cronk bomb that neither Tom Trbojevic nor Jorge Taufua wanted a part of in the 26th minute.
The Fijian scored his third and fourth in the space of just two minutes, taking another long ball from Munster in the 38th minute before finishing off a Cronk banana kick to give the Storm a 20-point halftime lead.
While Vunivalu scored the tries, Munster, Cronk and Dale Finucane were also immense for the Storm, who were able to rest captain Cameron Smith for the final 10 minutes of the match.
- AAP