Robert Roberson, 56, had been scheduled for the execution by lethal injection in Texas.
He was convicted in 2002 based on a since-discredited understanding of shaken baby syndrome.
Lawmakers in the Republican-controlled state House of Representatives have been reviewing Roberson's case as they debate whether to strengthen a Texas statute that addresses convictions linked to so-called junk science.
Roberson has said he found that his two-year-old daughter, Nikki, had fallen out of bed and, soon after, stopped breathing - days after a doctor diagnosed her with a viral infection and prescribed medicine that, according to medical experts, should not be given to young children.
In recent years, doctors have said that shaken baby syndrome abuse occurs and can cause symptoms that brain scans showed Nikki had, but that these also may be present in cases in which no abuse took place.
Roberson had repeatedly and unsuccessfully sought a new trial after Texas passed a 2013 law that gave convicted people a new avenue to appeal if their prosecution had turned on scientific evidence and hypotheses that had since become outdated or debunked.
On Wednesday, the lawmakers voted to issue a subpoena for Roberson to testify before the House Committee on Criminal Jurisprudence.
On Thursday, they secured an order from a Texas judge staying the execution and forbidding state officials from preventing Roberson from answering the subpoena.
Roberson is expected to appear in person at the state Capitol in Austin on Monday afternoon.
"This was an extraordinary manoeuvre by these Texas lawmakers, and it did the trick," said Robin Maher, director of the Washington-based Death Penalty Information Center.
"What I hope is that everyone involved in this case, particularly state officials pushing for a new execution date, will have an opportunity to reconsider their options."
