An official press release said a suspected pirate ship with two speed boats in tow approached the INS Tabar late Tuesday, threatened to blow up the warship, and opened fire. Indian naval officers saw well-armed men roaming the ship's deck.
CBS News correspondent Mark Phillips reports the crew of the Tabar said the pirate's main vessel was destroyed.
The Indian ship chased one of the speed boats, which was later found abandoned, while the other escaped.
The attack came the same day pirates hijacked a Thai boat and an Iranian bulk cargo carrier off Somalia's coast, and three days after pirates seized a Saudi supertanker.
There were also reports Wednesday morning that a Greek bulk carrier had been seized in the Gulf of Aden, off Somalia's coast. CBS News could not immediately verify the reports.
If confirmed, the Greek vessel would be the ninth ship to be seized in the area in the past two weeks, according to maritime officials.
Pirates hijacked the Thai fishing boat Tuesday with its 16 crew members in the Gulf of Aden, said Noel Choong, who heads the International Maritime Bureau's piracy reporting center in Kuala Lumpur. Also Tuesday, an Iranian bulk cargo carrier with 25 crew members was seized in the area, according to earlier reports.
Both vessels were heading to the Middle East when they were hijacked, he said.
The U.S. Navy's 5th Fleet based in Bahrain dispatched an aircraft to the area after the hijackings were reported, and spotted the two vessels in the hands of the pirates, 5th Fleet spokeswoman Cmdr Jane Campbell said.
She said there were no U.S. ships in the vicinity when the hijackings were taking place. In any case, ships would have to be "within 10 minutes responding time to prevent any hijacking," she told The Associated Press.
Choong said the Thai boat, which was flying a Kiribati flag but operated out of Thailand, made a distress call as it was being chased by pirates in two speedboats but the phone line got cut off midway. The bulk carrier was flying a Hong Kong flag but operated by the Islamic Republic of Iran Shipping Lines.
"There is no firm deterrent, that's why the pirate attacks are continuing. The criminal activities are flourishing because the risks are low and the rewards are extremely high," Choong said.
Kaj Larsen, a filmmaker who produced a documentary on pirates in southeast Asia, told CBS' The Early Show that hijacking a cargo ship is "extraordinarily easy."
Larsen said the only difference between the methods of pirates today compared with those of centuries past is the use of AK-47s and rocket-propelled grenades.
Piracy is a growth industry - Somalia's only one. In three years, the pirates have netted an estimated $30 million. For the hijackers, there are three sources of revenue from every ship seized; ransom for the crew, the cargo, and the ship itself which can be repainted, reflagged and resold.
CBS News correspondent Sheila MacVicar reports they've invested those profits and upgraded their gear. The marauders are now equipped with GPS, sophisticated communications equipment and rocket launchers. And now they're going after bigger prizes.
Choong said Tuesday that at least 17 vessels remained in the hands of pirates along with more than 300 crew members - including a Ukrainian ship loaded with arms and the Saudi Arabian supertanker carrying $100 million in crude that was taken over the weekend.
MacVicar reports that there is a specialist team on alert in Dubai, waiting for the pirates to make contact again from the massive Sirius Star tanker, but negotiations with pirates in control of a ship are often long and drawn out. The Ukrainian ship carrying Russian tanks and weapons that was seized in September is still being held by Somali pirates.
Tuesday's incidents brought the number of attacks in Somali waters this year to 95, with 39 ships hijacked. That number did not include the Greek vessel reportedly seized Wednesday.
Despite increased patrols by a multicoalition naval force, attacks have continued unabated off Somalia, which is caught up in an Islamic insurgency and has had no functioning government since 1991. Pirates have seized dozens of ships off Somalia's coast in the last year, generally releasing them after ransoms were paid.
NATO has three warships in the Gulf of Aden and the U.S. Navy's 5th Fleet also has ships in the region.
But, Campbell said, naval patrols simply cannot prevent attacks given the vastness of the sea and the high number of vessels passing through the Gulf of Aden - 21,000 every year.
"Given the size of the area and given the fact that we do not have naval assets - either ships or airplanes - to be everywhere with every single ship" it would be virtually impossible to prevent every attack, she said.
On Tuesday, a major Norwegian shipping group Odfjell SE ordered its more than 90 tankers to sail around Africa rather than use the Suez Canal after the seizure of the Saudi tanker, MV Sirius Star, on Saturday.
"We will no longer expose our crew to the risk of being hijacked and held for ransom by pirates in the Gulf of Aden," said Terje Storeng, Odfjell's president and chief executive.
The Gulf of Aden, off Somalia, connects to the Red Sea, which in turn is linked to the Mediterranean by the Suez Canal. The route is thousands of miles and many days shorter than going around the Cape of Good Hope off the southern tip of Africa.
Saudi Arabia, which is the world's leading oil producer, has condemned the hijacking and said it will join the international fight against piracy. Somali officials vowed to try to rescue Sirius Star by force if necessary.
The supertanker was anchored Tuesday close to Harardhere, the main pirates' den on the Somali coast, with a full load of 2 million barrels of oil and 25 crew members.