The twin-engine plane, bound for Minneapolis, reported the problem at 4:25 p.m. CDT, circled the airport to burn fuel and landed safely at 6:22 CDT.
Emergency crews rushed to the plane, which landed with no visible sparks.
Aerial video of the Embraer 145 plane showed the left tires on the landing gear nearly stripped of rubber.
The plane, which was en route to Minneapolis, tried to land once, but was waved off. It was ordered to do a second flyby of the airport so officials could assess the damage and allow the plane to burn fuel before it could land, said Continental Express spokeswoman Kristy Nicholas.
Continental spokeswoman Sarah Anthony said the plane carried 45 passengers and a crew of three.
"I want to commend the crew of flight 3161 for handling this difficult situation with the utmost in professionalism and ensuring the passengers safety," said ExpressJet President and CEO Jim Ream. ExpressJet Airlines Inc. operated the flight.
The passengers were taken by buses from the plane to the terminal about 30 minutes after the landing. They later boarded an 8 p.m. flight to Minneapolis, a flight that had been delayed an hour to accommodate them.
Julie King, a spokeswoman for Continental, said there were no injuries on board.
Airport spokesman Richard Fernandez said other airport operations continued normally as the plane circled overhead.
David Padovan, 30, from Houston, was supposed to fly from Minneapolis to Houston on the aircraft that had trouble.
He saw the news coverage of the blown-out tire on CNN, and said he felt "happy I wasn't the one on the plane."
Padovan said he hoped the passengers on board would land safely.
"Really, I was just kind of concerned," said Padovan, who was traveling from Wyoming to Houston and was in Minneapolis on a layover. "How are we going to get home?"
Padovan was moved to a 7 a.m. flight Wednesday, and was sent to a hotel for the night.
No injuries were reported in that incident.
An initial examination of the plane found one of the tires on the landing gear to be flat, said David Jackson, spokesman for American Eagle.