NASA and Roscosmos said search-and-rescue teams were in contact with the crew and en route to the landing location.
Bridenstine and the NASA team are monitoring the situation carefully, the space agency said, adding that NASA is working closely with Roscosmos.
"I fully anticipate that we will fly again on a Soyuz rocket and I have no reason to believe at this point that it will not be on schedule", the Nasa administrator said.
ISS operations integration manager Kenny Todd described the incident as a "major anomaly" but said he has "every confidence" that Russian space agency Roscosmos will establish the cause of the malfunction.
Still, Moscow has suspended all manned space launches until it confirms exactly what went wrong and why, and Rogozin has ordered a state commission to investigate.
The crew safely returned to Earth in a jettisoned escape capsule.
See the damage from massive Hurricane Michael in the Florida panhandle
After daylight Thursday, Florida residents took stock of the epic scale of the disaster. "I've' never seen anything like it". Michael is the third strongest hurricane to ever hit the U.S. mainland and the strongest to impact the Florida Panhandle.
Both Hague and Ovchinin will spend a few days at Star City for routine medical checks.
And with the news the crew - Alexey Ovchinin, a Russian cosmonaut, and Nick Hague an American astronaut - are in good health according to officials, attention has somewhat inevitably turned to the more humorous side of the misfire.
Yesterday's problem occurred when the first and second stages of a booster rocket, launched from the Soviet-era cosmodrome of Baikonur in the central Asian country, were separating, triggering emergency systems soon after launch.
The doomed booster left Earth behind at 4:40 AM ET today (Oct. 11), and everything seemed fine for the first several minutes.
This morning, the first launch since the possible sabotage was discovered, Russia's Soyuz booster saw its first in-flight failure in recent memory, and the first manned rocket-related emergency in decades. Hague and Ovchinin are out of the capsule and are reported to be in good condition.
He said the next unmanned cargo ship could go into space later than planned.
Bridenstine also heaped praise on the relationship Washington and Moscow enjoy in the frontier of space, free from the deepening political disputes "we have terrestrial". Despite its age, the Soyuz platform has been an extremely reliable mode of transportation to space.
In August, the International Space Station crew spotted a hole in a Russian Soyuz capsule docked to the orbiting outpost that caused a brief loss of air pressure before being patched.