The “extremely dangerous” storm is the strongest hurricane on modern record in the area, the US National Hurricane Center (NHC) says.
Grand Bahama residents have been evacuating from its predicted path.
A storm surge could be life-threatening and as high as 23ft (7m) in places, officials warn.
The storm made landfall at 12:40 (16:40 GMT) on Elbow Cay in the Abaco islands, northern Bahamas, the NHC said.
Local residents posted footage showing floodwaters engulfing some homes, their rooftops blown off.
Dorian, moving westward at 7mph, is expected to continue to move over Great Abaco Island and near, or over, Grand Bahama Island later on Sunday into Monday.
Authorities closed some airports in the outlying islands, but the main international airport remains open on Sunday.
After hitting the Bahamas, it should move closer to the Florida east coast late on Monday through Tuesday night local time.
US President Donald Trump said he was monitoring Dorian. He cancelled a planned trip to Poland, sending Vice-President Mike Pence in his place.
In a tweet on Sunday, Mr Trump warned Dorian was expected to hit “much” harder than originally thought, and urged residents in its path to be careful.
Hurricanes, which vary in strength from category one to five on the Saffir-Simpson scale, tend to get stronger as they move over warm waters like those off the coast of Florida.
Dorian has also grown larger in size, with hurricane-force winds currently extending 45 miles (75km) from the storm’s centre.
How has the Bahamas been preparing?
Prime Minister Hubert Minnis earlier announced an evacuation order for parts of Grand Bahama and the Abaco Islands, both in the north of the archipelago. All tourists were asked to leave the islands.
At a news conference, Mr Minnis begged residents to head for the main island to escape the “devastating, dangerous storm”.
“I want you to remember: homes, houses, structures can be replaced. Lives cannot be replaced,” he said, adding that 73,000 people and 21,000 homes were in danger.
How destructive are category five hurricanes?
With sustained wind speeds of at least 157mph they are relatively rare, although four other storms reached this level in the last three years:
- Michael made landfall in Florida last year, causing 59 deaths and about $25bn (£20bn) in damage. It was the strongest storm to make landfall in the US in 26 years and only the fourth hurricane of this category to have hit the country
- Maria, in 2017, devastated the island of Dominica, where it was responsible for 31 deaths, before causing widespread destruction in Puerto Rico as a category four hurricane. A study said nearly 3,000 people had died there as a result
- Irma, also in 2017, made seven landfalls, four of which were category five, across the northern Caribbean Islands, before making landfall in the Florida Keys as a category four storm. It was one of the strongest and costliest hurricanes on record and caused 47 deaths in total
- Matthew, in 2016, was a category five storm that later made landfalls in Haiti, Cuba and Grand Bahama Island as a major hurricane. It was the deadliest hurricane in more than a decade, with 585 deaths, more than 500 of them in Haiti alone
Source: US National Hurricane Center
What about the US?
Florida was earlier expected to take a direct hit in the storm. Its governor, Ron DeSantis, has activated 2,500 National Guard troops and has another 1,500 on standby.
As Dorian crawled towards the US, shoppers were seen queuing around blocks to buy supplies such as medication and fuel.
Popular attractions, including Disney World in Orlando, have said they are “closely monitoring” the storm. The coastal city of Miami has also ordered the removal of electric rental scooters from the streets to avoid them becoming projectiles.
Palm Beach and Martin counties issued mandatory evacuations for some residents, including for those in mobile homes, barrier islands and low-lying areas.