The ship left from Cape Liberty, New Jersey, Saturday for a seven-night cruise to the Bahamas.
Video taken on board the ship shows big, blue waves tumbling alongside the vessel, along with overturned plant pots and chairs and other cosmetic damage.
Petty Officer Mark Barney, of the public affairs office of the U.S. Coast Guard, said that coast guard officials contacted the captain of Anthem of the Seas on Sunday knowing they were in heavy seas and after seeing social media posts.
Sunday, as the ship powered into a strong winter storm, the captain used the ship's intercom to order passengers confined to their cabins and tell them his crew had underestimated the storm's strength.
A buoy in the Atlantic about 400km south of Cape Hatteras reported wave heights of nine meters and wind gusts of 118km/h late Sunday, according to the National Oceanic and Atmospheric Administration.
In a statement, a Royal Caribbean spokeswoman said four passengers were injured during the storm, none seriously.
Guests will get a full refund and a certificate toward a future cruise.
Passengers report being able to leave their cabins early Monday morning.
Royal Caribbean released a statement saying the ship experienced "extreme wind and sea conditions, with wind speeds higher than what was forecasted". "The Captain regularly gave us updates".
The Coast Guard told CBS News the ship "diverted course as a precautionary measure" but wasn't in distress, adding that it was in touch with the Royal Caribbean every hour, and that the weather was improving.
The ship was on its way from the New York City area to Port Canaveral, Florida, when it not only encountered the storm, but really sailed right into the heart of it.
Through there was damage to some public areas and guest rooms, Royal Caribbean said it hasn't affected "the seaworthiness of the ship".
In a sales point for satellite data and television service providers, the severe storm did not interrupt the ships' reception of Sunday's Superbowl coverage, nor its internet connection.
Passenger Gabriella Lairson said via Facebook on Monday that the storm started about 3 p.m. Sunday.
Robert Huschka, the editor of the Daily Echo's sister paper, the Detroit Free Press, is on board the ship.
Royal later tweeted an apology to its passengers: "We appreciate our guests' patience and cooperation".
The Royal Caribbean ship Anthem of the Seas, the third largest cruise ship ever built, is making its way back to New Jersey after the storm shattered glass and toppled furniture, injuring four people. "While the weather was unpleasant, the ship remained seaworthy at all times".
MS Anthem of the Seas was travelling from New Jersey to Florida off the East Coast of the United States when it got caught in the extreme storm.
"I'm not going to lie: I was terrified, although I did my best to hide it from my wife".
"The waves out there looked as if they were taller than the ship", Garraffo said.