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.
Caption: NASA tracks storm from ISS as Hurricane Dorian barrels into north-west Bahamas