The Beazley family's prayers were answered, for now
The stay came after a 136-page appeal was filed by Beazley's lawyers hours before he was due to die at 1800 (2300 GMT) on Wednesday.
The case of Beazley, who is now 25, but committed murder at the age of 17, has reignited international criticism of capital punishment in the United States.
Officials at Huntsville State Prison said Beazley smiled and seemed barely able to take in the news before saying: "I'm fine, I just have to comprehend this, give me a second."
Appeal pending
The Texas Court of Criminal Appeals decided to consider his lawyers' arguments questioning the legality of executing someone who had committed murder as a minor.
Beazley still has an appeal before the Supreme Court seeking a full review of his case.
This is expected to hinge on whether the execution of people who were under the age of 18 when they committed their crimes is constitutional.
In 1989, a Kentucky high court ruled that the constitution did allow death sentences for defendants as young as 16.
The case could also come before the Governor of Texas, Rick Perry, who has refused to halt any of the 11 other executions in the state since he took office last year.
Beazley's lawyer, Walter Long, said: "I'm so grateful that Napoleon is still alive, grateful for his family... and I'm also grateful that our really critical legal issues remain alive."
However the lead prosecutor in the case, District Attorney Jake Skeen, vowed to fight on, insisting the punishment must be carried out in the name of justice.
"We still hold the execution is proper and the just sentence, and we will continue to seek and strive to see the imposition of this sentence," he said.
Botched carjacking
Beazley shot 63-year-old John Luttig, a prominent businessman in the driveway of his home in Tyler, Texas seven years ago in a botched attempt to steal his car.
Two accomplices received jail terms of 40 years, but Beazley was sentenced to death.
As well as pointing to the killer's age, critics of the death sentence argue that Beazley did not get a fair trial as Mr Luttig's son, Michael, now sits as an appeals court judge in Richmond, Virginia.
Three of the nine US Supreme Court justices excused themselves from the appeal because of their ties to Michael Luttig, who had either worked for them or advised them.
There have also been allegations that Beazley's trial was prejudiced because he is black, and that his lawyer was incompetent, failing to mention his client's age at the sentencing hearing.
The case has wider implications, being just the latest in the state of Texas, which carried out a record number of executions - 40 - last year.
Since the US Supreme Court allowed states to reinstate the death penalty in 1976, 18 people who committed murder under the age of 18 have been executed across the United States.