The deal, first reported by the Atlanta Journal-Constitution, was confirmed by both teams later Saturday night.
The Padres are also sending $10.5 million to the Braves as part of the trade.
A source confirmed to ESPN's Jerry Crasnick that Olivera is expected to be designated for assignment by San Diego on Tuesday, when he comes off the restricted list following an 82-game suspension for violating Major League Baseball's domestic-violence policy.
The suspension, which is retroactive to April 30, stemmed from Olivera's April 13 arrest on assault and battery charges at a hotel outside Washington, D.C.
Kemp signed an eight-year, $160 million deal with the Dodgers before the 2012 season. He has a $21.5 million salary this year and is owed the same amount in each of the next three seasons.
"We felt like it's just a situation where we can use that money down the road," Padres general manager AJ Preller said. "Reallocate to our bigger club, reallocate it potentially to the amateur market, which is a spot where there's a lot of value in the flexibility with the dollars presented to us.''
Olivera agreed in early 2015 to a $62.5 million, six-year contract with the Los Angeles Dodgers, then was traded to the Braves last July. He has a $4 million salary this year but lost $1,792,350 because of his suspension. He is owed $6 million next season, $6.5 million in 2018, $7.5 million in 2019 and $8.5 million in 2020.
Olivera is on a rehab assignment with Triple-A Gwinnett but was removed from the lineup before Saturday night's game. He is hitting .211 with no homers and two RBIs in six games this season.
San Diego acquired Kemp, a two-time All-Star outfielder with the Dodgers, from Los Angeles in December 2014. It took several days to consummate the trade because Kemp had to be cleared medically.
Despite arthritis in both hips, Kemp, 31, could boost the weakest offense in the major leagues. Atlanta has baseball's worst record and ranks last in runs scored and homers.
In 254 games with the Padres, Kemp is hitting .264, 28 points lower than his nine-year batting average with the Dodgers. While with San Diego, he recorded 46 homers, 169 RBIs and 247 strikeouts.
"It's a sign that we aren't happy losing," Braves general manager John Coppolella said of acquiring Kemp. "We aren't trying to finish with the worst record ever so we can get the first [draft] pick. We want to win every night, and making this trade now is a big deal to us because it helps us get better this year and the next three years after that."
Kemp is expected to make his Atlanta debut on Tuesday, when the Braves begin a three-game home series with Pittsburgh. President of baseball operations John Hart said there was no need for Kemp to hurry to Turner Field; the Braves have an off day Monday.
Atlanta had tried for several months to deal Olivera following his April arrest. Coppolella said he was pleased to put the Olivera chapter behind him. Saturday marked the one-year anniversary of the third baseman's arrival.
"What happened with Hector is unfortunate and unacceptable," Coppolella told the AP. "We're happy to have Matt Kemp."
San Diego is sending Atlanta $3 million this year as part of the trade: half on Aug. 15 and the rest on Sept. 15. From 2017 to '19, the Padres will pay the Braves $2.5 million annually, half each on May 15 and July 15.
"To get him for roughly $8.5 million a year when you factor out Olivera, it made a lot of sense for us," Coppolella said of Kemp. "We didn't give up a draft pick. We didn't trade away any prospects. So we were able to stick with the plan that we had while still filling a major need for right-handed power."
Saturday's trade is the latest among a series of deals the Padres have made in the past two months as they move to rebuild.
Since June, they have traded away four starting pitchers, sending All-Star left-hander Drew Pomeranz to the Boston Red Sox, James Shields to the Chicago White Sox, and Andrew Cashner and Colin Rea to the Miami Marlins.
Atlanta also has taken on bad contracts for declining players such as Michael Bourn, Nick Swisher and Bronson Arroyo. Dan Uggla was released in July 2014 despite the Braves still owing him $18 million.
Information from ESPN's Jim Bowden and the Associated Press was used in this report.