Among the injured is the USS Fitzgerald's commanding officer, who was taken to hospital by helicopter, the BBC reported.
The destroyer collided with the container ship about 56 nautical miles (104km) south-west of Yokosuka.
Aerial images showed heavy damage to the Navy ship's starboard side after the collision at about 02:30 local time on Saturday (17:30 GMT Friday).
The ship's commanding officer, Cdr Bryce Benson, was in a stable condition after being transferred to U.S. Naval Hospital Yokosuka, according to Navy officials.
Two other sailors were flown to hospital by helicopter for treatments for "lacerations and bruises", said the U.S. 7th Fleet via Twitter.
By mid-afternoon local time, it said that all flooding on board the USS Fitzgerald was under control. It is not believed to be at risk of sinking.
The 7th Fleet also said the 154-metre (505ft) ship was now on its way to Yokosuka – under its own power, but with limited propulsion.
Read alsoUSS Oscar Austin enters Black Sea (video)The area where the collision took place is extremely busy, with shipping heading in and out of Tokyo bay.
However, the USS Fitzgerald is one of the most advanced warships in the world – with highly sophisticated radars systems.
At the moment the focus is on the search for the missing sailors. Later will come the investigation into how one of the world's most sophisticated warships failed to avoid colliding with a 30,000-tonne cargo ship on a calm, clear night.
The coastguard said the ACX Crystal container ship carried a Filipino flag and weighed just under 30,000 tonnes, about three times that of the USS Fitzgerald.
It sustained lighter damage to the port bow, according to Japanese national broadcaster NHK.
The Associated Press agency said there were no injuries reported on board the ACX Crystal, a 222-metre (730ft) vessel.
In a statement released on social media, U.S. Navy operations chief Adm John Richardson said: "As more information is learnt we will be sure to share it with the Fitzgerald families and when appropriate the public. [...]
All our thoughts and concerns are with the Fitzgerald crew and their families."