The pirates, who often demand high ransom payments, hijacked the tanker's crew of five South Koreans and 19 Filipinos when the vessel was about 1,560km east of the Somali coast.
A Nairobi-based maritime group said a pirate source identified only as Mohamed said the ship was now heading for Haradheere, the port and pirates' base where many ships are held during ransom negotiations.
Kim Young-sun, the foreign ministry spokesman, said the aim was to "ensure the safety of the crewmen and the success of possible negotiations".
The ministry said the navy received a call from the Samho Dream saying three Somali pirates had boarded the vessel.
The destroyer can travel much faster than the supertanker and would be able to reach the ship before it reaches any port.
Valero Energy Corp, a Texas-based refiner, said it was the owner of the crude oil and that the cargo was bound for the US Gulf Coast.
"Asian importers including Korean will see little impact as their crude oil imports from the Middle East are moving in the opposite direction. US importers will have problems," said a trader at the country's biggest oil refiner, SK Energy.
Frequent hijackings have upped insurance costs, forced some ships to go around South Africa instead of through the Suez Canal.
Although attacks in the Gulf of Aden and the Indian Ocean are commons, it is rare for pirates to successfully seize the kind of massive supertankers that carry most Gulf crude to refiners.
The first successful assault on a very large crude carrier (VLCC) occurred in late 2008 when pirates took the Saudi-owned Sirius Star,.
Another, the Greek-flagged Maran Centaurus, was seized last November and held for nearly two months until a ransom, believed to be between $5.5m and $7m, was paid.
Globally in 2009, there were 406 reported incidents, in which 153 vessels were boarded and 49 were hijacked.
There were 84 attempted attacks and 120 vessels were fired on.
A total of 1,052 crew members were taken hostage and at least 68 crew members were injured while eight were killed.
In all, Somali pirates were held responsible for 217 acts of piracy in 2009, in which 47 vessels were hijacked and 867 crew members taken hostage.