"Congratulations on reaching orbit on the first attempt!" Musk wrote in a post to Bezos on X.
Dave Limp, CEO of Bezos's space company Blue Origin, said he was "incredibly proud".
"We'll learn a lot from today and try again at our next launch this spring," he added.
Bezos's team overcame technical barriers that caused delays earlier this week when ice formation halted a launch.
Blue Origin's employees and crowds gathered near Cape Canaveral cheered as the 98 meters-high rocket hurtled into orbit.
But the company failed to land New Glenn's main rocket engine, or booster, onto a platform in the Atlantic Ocean.
It had hoped that the booster would be reusable for future launches but after about 20 minutes of flight, the company confirmed it had lost the engine.
Bezos's company Blue Origin has struggled to match the pace set by SpaceX. But this launch will be seen as a major step forward for the business.
The New Glenn rocket was named after John Glenn, the first American astronaut to orbit Earth more than 60 years ago.
The rocket is more powerful than SpaceX's most commonly-used rocket, the Falcon 9. It can also carry more satellites, and Bezos wants to use it as part of his Project Kuiper, which aims to deploy thousands of low-earth satellites to provide broadband services.
That project would compete directly with Musk's Starlink service.