The teams agreed to the trade last week during the winter meetings in San Diego. It was the first of three big deals by new Padres general manager A.J. Preller, who also has added outfielder Wil Myers, the 2013 AL Rookie of the Year, from Tampa Bay and All-Star catcher Derek Norris from Oakland.
The Padres, desperate to pump up the worst offense in the majors, also get $32 million from the Dodgers to help offset the $107 million remaining on Kemp's contract. The Padres' obligation of $75 million to Kemp over five years becomes the biggest deal in club history.
Kemp twice made the All-Star team during his time in L.A., as well as winning two Gold Glove and two Silver Slugger awards. His 182 homers rank fourth in Los Angeles Dodgers history.
Kemp weighed in on the deal via social media, penning a "goodbye to the city and the fans who have been there since the beginning of my career."
Padres president Mike Dee declined to comment on reports that the team was concerned with certain aspects of Kemp's physical.
"It was a process that played out in a way that was, I think, consistent with what we thought going into it," Dee told The Associated Press. "We had a player that had some prior injures and we wanted to make sure we did a full exam. A.J. and his team did their due diligence, conferred with our medical team and ultimately were comfortable moving forward."
The deal was completed despite some possible lingering ill will between the Dodgers' front office and Preller because of the belief that Kemp's physical results were leaked to help San Diego pry further concessions out of its NL West foes, sources told ESPN.com's Ramona Shelburne. The Padres did ask for additional compensation from the Dodgers after Kemp's physical, but Los Angeles refused.
The deal was cast into some doubt after USA Today reported Thursday that Kemp's physical with the Padres revealed arthritis in both hips. The Dodgers were upset that Kemp's medical information leaked publicly, sources said, though one Dodgers source told ESPNLosAngeles.com's Mark Saxon that the team never felt the deal was seriously jeopardized.
Former Dodgers scouting director Logan White, who now works in San Diego, was viewed as one of the driving forces behind the deal's completion.
Kemp has had shoulder and ankle operations in recent years. Last May, he was groomed to play left field because the Dodgers weren't confident he could handle the defensive demands of playing center field after undergoing ankle surgery the previous offseason.
The Padres had a historically awful offense in the first half of 2014 before pulling out of it to finish third in the NL West. Still, they were at the bottom of the majors with a .226 average, .292 on-base percentage and 535 runs. Their 109 homers were the second-fewest in the bigs.
The 30-year-old Kemp had a strong second half in 2014 for the NL West champion Dodgers. He finished the season with a .287 average, 25 home runs and 89 RBIs, hitting 17 of those homers after the All-Star break while driving in 54 runs. He led the major leagues in slugging percentage during that span.
Kemp was the runner-up for the National League MVP in 2011, but he has dealt with a long string of injuries since April 2012. He underwent two shoulder surgeries and a major operation on his right ankle and had three hamstring strains. Until now, he had never been diagnosed with a hip ailment.
The Dodgers had been looking to trade an expensive veteran outfielder -- either Kemp, Andre Ethier or Carl Crawford -- for more than a year. They still will listen to offers for Ethier and Crawford since they are intent on giving prospect Joc Pederson an everyday opportunity and are content with Scott Van Slyke as their fourth outfielder.
Grandal, 26, led the Padres with 15 homers last season while hitting .225.
He was suspended for the first 50 games of 2013 after testing positive for testosterone and then tore up his right knee in a collision at the plate that July. He's also played first base.
Also Thursday, the Padres acquired Norris, right-hander Seth Streich and an international signing slot from the Oakland Athletics for right-handers Jesse Hahn and R.J. Alvarez. They also are nearing a deal for right-hander Josh Johnson, and have agreed to an 11-player, three-team trade that will bring them Myers in a swap completed Friday afternoon.
Norris, 25, batted .270 with 10 home runs and 55 RBIs in 127 games for the A's, who lost the wild-card game at Kansas City.
Information from ESPNLosAngeles.com's Mark Saxon and The Associated Press was used in this report.