It will be Japanese billionaire Yusaku Maezawa, the founder and chief executive of online fashion retailer Zozo.
Musk said that he will be flying on the company's future massive rocket, dubbed the "Big Falcon Rocket" (BFR), which is scheduled to be launched somewhere around 2023.
He revealed he purchased all available seats for the BFR lunar mission and plans to invite six to eight artists from around the world to go with him.
"Finally I can tell you that I choose to go to the moon", said Maezawa.
Before he introduced Maezawa, Musk outlined his company's space travel goals, warning of a possible event that could "end civilization", adding, "We should take action and become a multi-planet civilization as soon as possible".
Whoever the mystery passenger is will be among the first space tourists to fly with a commercial company.
SpaceX CEO Elon Musk revealed him to the world on Monday night after days of speculation about who the passenger might be.
SpaceX gave no details of the flight but said it would be "an important step toward enabling access for everyday people who dream of traveling to space".
Neither Maezawa nor Musk would disclose the price of the flight, but Musk said that Maezawa - who had been one of the two customers for SpaceX's earlier plans for a circumlunar flight on a Dragon announced almost 18 months ago, but since shelved - had already made a down payment.
Musk said Monday that the total development cost of the rocket is expected to be around $5 billion. SpaceX's crewed missions are now scheduled to begin by the middle of next year.
He wrote on his caption: 'My mission to go around the moon, as the first private passenger, will represent the imagination and curiosity of all people.
The current record was set on 15 April 1970 by the three-person crew of Apollo 13, who made it 400,171 km (248,655 mi) from Earth as their crippled craft swung around the dark side of the moon. The BFR ship won't be making a stopover on the lunar surface, though.
Musk says the BFR will act more like a sky diver than an airplane.
The BFR will eventually replace Falcon 9 and Falcon Heavy, but not until all of SpaceX's customers are on board, SpaceX President Gwynne Shotwell said in April.
During the announcement last year, SpaceX noted that, "We are excited to announce that SpaceX has been approached to fly two private citizens on a trip around the moon late next year", with the company further noting that the two individuals had already paid a significant deposit for the moon mission.
It'll have forward actuated fins, and rear actuated fins, and it'll be powered by SpaceX's Raptor thrust engine.