After the flight, the closely held company said SpaceShip Two had climbed above 271,000 feet, or about 51.4 miles, reaching a maximum speed of 2.9 times the speed of sound.

The U.S. Air Force and the National Aeronautics and Space Administration consider 50 miles up to be the edge of space, though some scientists, space buffs and international record-keeping authorities say space starts even higher.

Virgin Galactic years ago led the way in sparking interest in blasting tourists and small satellites into space using cheap rockets and various other unconventional launch systems. Experts consider it the fastest-growing segment in commercial space. But a December 2014 accident caused Virgin Galactic to assume a significantly lower profile.

Now, the company has bounced back from the crash that killed a co-pilot and prompted federal criticism of design principles and what investigators determined was inadequate consideration of potential human error in pilot training.

Thursday’s flight could put Virgin Galactic in a strong position to compete with Blue Origin, a rival space-tourism startup run by Amazon.com Inc. founder Jeff Bezos. The ventures are vying to be first to carry paying passengers outside the atmosphere in a U.S.-built spacecraft. Both companies have indicated commercial operations are likely to start sometime next year, but schedules are fluid and the outcome of future test flights could alter those plans. Mr. Branson’s ambitions also include launching small satellites using a rocket released from a modified Boeing Co. 747.

Over the years, Mr. Branson has sought more than $1.4 billion in funding from Middle Eastern investors for his space-plane venture, but much of that support is now uncertain due to Virgin Galactic’s decision to sever financial ties with Saudi Arabia.

Mr. Branson and his team originally rolled out an early version of the rocket-powered craft nine years ago and initially hoped to begin regular flights several years later.

During the intervening years, Virgin Galactic brought manufacturing and testing of its winged spacecraft in-house while revamping safety and quality-control safeguards. The 2014 flight was touted as a record-setting event before the accident. Since then, Mr. Branson has refused to predict test flight schedules.

Virgin Galactic on Wednesday posted a message on Twitter seemingly aimed at tamping down expectations, suggesting the rocket motor perhaps wouldn’t be ignited for the full duration needed to travel out of the atmosphere.

But after the flight, the company posted a flurry of tweets on the altitude and speed of SpaceShip Two.

Virgin Galactic’s milestone also amounts to vindication for the billionaire entrepreneur known for his marketing savvy and personal commitment to space exploration. In the past, Mr. Branson has said he and some family members expect to be on the first one of his space planes carrying passengers.

Mr. Branson released a statement saying “it was an indescribable feeling: joy, relief, exhilaration and anticipation for what is yet to come.”

While flying in clear skies Thursday in California with SpaceShip Two tucked under its wings, the four-engine carrier plane dubbed White Knight Two released the mostly white spacecraft at around 8 a.m. local time. The rocket motor fired as expected for 60 seconds, blasting the test vehicle into a suborbital trajectory. It deployed a movable tail to slow down during the descent and glided back toward its base, landing like a conventional aircraft about 45 minutes after takeoff.

Thursday was the fourth time the spacecraft, also called VSS Unity, has flown under its own power, with the rocket motor firing longer than in previous flights. The company said Wednesday that the test was intended to obtain data on supersonic-handling qualities and thermal dynamics. That ground controllers gave the green light for full-duration thrust indicates the vehicle didn’t exhibit any unexpected problems or handling difficulties.

The flight marked the first time since the retirement of NASA’s space shuttle fleet in 2011 that a U.S.-built vehicle carried people into space. American astronauts travel to and from the international space station aboard Russian-built rockets and capsules.

Boeing Co. and Elon Musk’s Space Exploration Technologies Corp. next year plan to start transporting astronauts into orbit on separate, domestically produced rockets and capsules. Both companies have longer-term plans for carrying tourists into space.

Following Thursday’s landing, a jubilant Mr. Branson shed tears of joy as he hugged bystanders, reflected on 14 years of effort and ruminated about the future. No other private spacecraft specifically designed to carry paying passengers has ever flown to the boundary to space.

George Whitesides, the company’s chief executive and a former senior NASA official, said reaching the elusive goal amounted to “more compelling evidence that commercial space is set to become one of the 21st century’s defining industries.”

The test mission, which carried several experiments on behalf of NASA, allowed Virgin Galactic to say it marked the first commercial revenue from a company flight.

Virgin Galactic is building additional space planes. But industry experts have said ramping up production, hiring the necessary expertise and maintaining robust launch rates pose huge challenges for Virgin Galactic, which so far has focused on low-volume, boutique production practices.

Write to Andy Pasztor at andy.pasztor@wsj.com