Ike also hit a large stretch of coastline in neighbouring Louisiana.
Texas Governor Rick Perry said: "We pre-positioned the largest search and rescue operation in the history of the state of Texas in advance of this storm.
"There are thousands of personnel and vehicles that were very close in to where the storm struck the coast."
'Not worst-case scenario'
He said rescuers were moving into Galveston by air, by boat, and by ground.
See the likely path of Hurricane Ike through the US
About 140,000 people in the town and other low-lying areas chose to stay despite warning they could die, officials say.
The hurricane left Galveston flooded and without water or electricity.
But officials were encouraged by the fact that that the storm surge turned out to be much lower than forecast - 13.5ft (4m) after predictions of 20 to 25ft foot (6 to 7.5m).
Sedonia Owen, 75, and her son were among those residents who decided to stay in Galveston to protect their neighbourhood from possible looters.
"My neighbours told me, 'You've got my permission. Anybody who goes into my house, you can shoot them," Mrs Owen was quoted as saying by the Associated Press news agency.
Hurricane Ike caused widespread destruction in Galveston, Texas
Ike also damaged Houston, the fourth-largest city in the US.
A BBC correspondent who weathered the storm there described how ferocious winds ripped the glass from many of the city's skyscrapers.
The two people known to have been killed by the storm are a woman in Pinehurst, Texas, and a teenage boy in Louisiana's Bayou Dularge, according to the Associated Press.
Ike also left millions of people in Texas and Louisiana without power.
Production was suspended at 14 oil refineries and 28 natural gas processing plants that were in the storm's path.
However, US Homeland Security Secretary Michael Chertoff later said that overall "the impact... was not quite as bad as the worst-case scenario, it was still substantial".
Ike was downgraded to a tropical storm by the US National Hurricane Center at 1900 GMT, as it moved inland with top sustained winds of nearly 60mph (95km/h).
It is moving towards north-east Texas and is expected to reach western Arkansas in the coming hours.
Earlier, Ike caused devastation in Cuba and Haiti, where hundreds of people have died in several tropical storms over the last month.
Are you in the areas affected? Are preparing to evacuate or are you staying in your home? Send us your comments and experiences using the form below.
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