The militants, who call themselves Sinai Province, posted pictures online of what looked like a missile followed by a large explosion on the ship.
Egyptian officials said a coastguard vessel caught fire after a fire-fight with militants on the shore.
The army said there were no casualties, contradicting the militants' claim.
The incident happened in waters off the north Sinai town of Rafah, which neighbours the Gaza Strip.
The SITE Intelligence Group carried a statement and pictures from Sinai Province which it said showed a guided anti-tank missile moments before it struck the vessel, and then the moment of impact.
Witnesses said they saw other naval boats rushing to the scene to rescue those on board the burning vessel.
It follows a wave of attacks on the Egyptian military by Sinai Province.
Earlier this month, more than 100 people - including at least 17 soldiers - were killed in clashes after militants  launched near-simultaneous raids on military checkpoints in Sheikh Zuweid and Rafah in Sinai.
Analysis: BBC security correspondent Frank Gardner
Not surprisingly, there is a wide gap between the two versions reporting the maritime attack off the north coast of Sinai.
IS - specifically their local affiliate called "Sinai province" - claim they fired a shore-to-ship missile at the craft and support this with photos on social media showing a massive fireball that would certainly have killed many crew and possibly destroyed the whole ship.
Egypt officially denies there were any casualties, saying there was a fire-fight during which a coastguard boat was set on fire.
Since anyone found contradicting official casualty figures there risks a $60,000 fine, it is likely that the truth lies somewhere between the two versions.
Either way, this is a worrying development for shipping in the region. It comes just two weeks after a major battle between Egyptian forces and IS in Sinai, and it's clearly proving difficult to remove the militants and their growing arsenal from the area.
Egypt has vowed to rid its Sinai Peninsula of the militants, and has had a state of emergency and a curfew in place since last October.
Sinai Province was previously called Ansar Beit al-Maqdis (Champions of Jerusalem), but announced a name change in November 2014 when it pledged allegiance to IS, the militant organisation that has taken over large parts of Iraq and Syria.
The group has been on the rise since the military overthrew Islamist President Mohammed Morsi in July 2013, and has killed at least 600 police and armed forces personnel since then.
Jihadist attacks on shipping in the Middle East
Islamic State-affiliated militants in Sinai say they have carried out a missile attack on an Egyptian naval vessel in the Mediterranean Sea.