The photo was shared by Timothy McIntosh, who told the paper that his mother-in-law, Trudy Lampson, owned the nursing home.
His wife, Kimberly, said she had spoken to her mother on Sunday morning local time.
"She said it was a disaster and they were hoping the National Guard would come," she said.
The photo, which had been retweeted more than 3700 times, caused consternation on social media, where users appealed to emergency services to rescue them.
McIntosh, whose photo led some to question its authenticity, later confirmed they had been rescued and thanked the rescuers.
"Thanks to all the true believers that re-tweeted and got the news organizations involved," he added. "It pushed La Vita Bella to #1 on the priority list."
Harvey came ashore over the weekend as the most powerful hurricane to hit Texas in more than 50 years and has killed at least two people.
The death toll is expected to rise as the storm triggers additional tidal surges and tornadoes, with parts of the region expected to see a year's worth of rainfall in the space of a week.
* This article originally appeared on the Daily Telegraph and is reproduced with permission.
