The tiniest victim of a devastating Thanksgiving blaze in Brooklyn was clinging to life yesterday, and investigators described how a botched attempt to put out a mattress fire likely led to the horror.

Eight-month-old Josiah Alexis was placed on a ventilator along with his mom, Justine, 20, Justine’s brother Jonathan, 22, and her brother-in-law, George DeJesus, at New York Presbyterian Cornell Weill Medical Center in the aftermath of the fire, which sources have said was sparked by someone smoking in bed at the building on President Street in Park Slope.

All of them were in critical condition, as was Josiah’s other aunt — Justine’s twin sister, Jasmine, who is married to DeJesus — at Jacobi Medical Center in The Bronx, officials said.

One hero firefighter said it was nothing short of a miracle that rescuers reached little Josiah and the other victims as quickly as they did — responding in just 2 minutes and 27 seconds.

“The baby was limp, not breathing, in cardiac arrest,” said Neil Malone of Engine Company 239.

He said he gave the baby the “breath of life” and was doing chest compressions “using my thumbs.”

Josiah began gagging, Malone said, and “it was like a song to your ears when the baby gets a breath on his own.”

“As chaotic as the scene was, it was unbelievable how it all went so smoothly,” Malone recalled of pulling out the tiny victim.

“I’ve been in a lot of fires, but to find that baby in all of that debris is amazing.

“I’m hoping the family pulls through. I’m thankful they all survived.”

The tragedy stemmed at least partly from futile attempts to extinguish a mattress that caught fire from the cigarette, authorities noted.

“One of the people dragged the mattress through other rooms in an attempt to get it into the bathroom and extinguish the fire,” a source told The Post.

“But instead, the fire spread and trapped everyone inside.”

Homeowner Serge Alexis, who is Josiah’s great-grandfather — was overwhelmed with grief.

“I touched him — I tried to wake him up,” Alexis said of a visit Thursday night to his great-grandson. “He was a smiling baby, always smiling.”

Magaly Elie, the baby’s godmother and aunt of Justine and Jasmine, said the family is now praying for the baby’s survival.

“The baby is clinging to his life,” she said. “It’s hard for him to breathe.”

Authorities said they won’t know for sure how the fire started until they are able to interview the adults badly hurt in the blaze.

“As the victims become available to be interviewed, we’ll have a better idea,” a source said.