Leno, 62, will wrap up what will be 22 years as host of "The Tonight Show" in the spring of 2014 - some seven months before his contract was officially due to end.
Fallon, 38, the current host of "Late Night with Jimmy Fallon" will "transition into new hosting duties on 'The Tonight Show'" after Leno ends his run, NBC said in a statement.
The network also said it was moving "The Tonight Show" from its Burbank studio, outside Los Angeles, to New York, where it began in 1954.
No specific date was announced, but the change will take place in conjunction with NBC's broadcasts of the Winter Olympics in Sochi, Russia, next February.
Wednesday's announcement ended months of speculation and followed a skit on Fallon's show by him and Leno on Monday night in which they played down reports of rivalry and made fun of the rumor mill.
Leno, who was replaced by Conan O'Brien in 2009, only to return a few months later in a public relations debacle for the network, congratulated Fallon.
"I hope you're as lucky as me and hold on to the job until you're the old guy," he said in a statement.
Fallon, who started out on the long-running "Saturday Night Live" comedy show in 1998, said, "I'm really excited to host a show that starts today instead of tomorrow," referring to his current program's post-midnight start time.
"We are purposefully making this change when Jay is number-one, just as Jay replaced Johnny Carson when he was number-one," said Steve Burke, NBCUniversal's CEO.
"The Tonight Show" has maintained a hold on U.S. popular culture for decades, offering a forum for celebrities to promote their latest ventures and a springboard to fame for many standup comedians.
The program currently leads its three late-night rivals in overall audience, attracting about 3.5 million viewers, compared with about 3 million for CBS rival David Letterman.
But the average age of viewers for Leno and Letterman, 65, is in the mid-50s - higher than the 18-49 demographic preferred by advertisers.
ABC upped the stakes in January by moving Jimmy Kimmel, 45, to the late-night slot in a bid to grab a younger audience. Kimmel's ratings have challenged both Letterman and Leno in the 18-34 age group, while his overall audience is about 2.6 million, according to the most recent Nielsen data.
Kimmel proffered a winking posting via Twitter on Wednesday, saying, "congratulations to my dear, sweet @jimmyfallon - a formidable rival and an incredible lover."
NBC said "Saturday Night Live" and "30 Rock" producer Lorne Michaels would serve as executive producer of the relocated show. It will be broadcast from NBC headquarters in New York's Rockefeller Center.
New York Mayor Michael Bloomberg and the state's governor, Andrew Cuomo, welcomed the show's return to New York.
"We couldn't be happier that one of New York's own is bringing the show back to where it started - and where it belongs," Bloomberg said in a statement referring to Fallon's Brooklyn roots.
Carson, who hosted the program from 1962 to 1992, moved the show to Southern California in 1972.
NBC said that programming plans for the 12:35 a.m. time slot now filled by Fallon's show would be announced soon.
NBC is a unit of Comcast Corp , ABC is a unit of Walt Disney Co and CBS is part of CBS Corp .