Wilson's deal is worth $1 million and also includes performance bonuses, a team source told ESPN's Jim Bowden.
Wilson, 31, underwent Tommy John surgery last April after two early-season appearances with the Giants. He was non-tendered by the club last November after making $8.5 million in 2012.
The tentative plan is for Wilson to begin a two-week program to return to the majors starting at the Dodgers' spring training complex in Arizona, according to reports. If all goes well, he will continue on to Class A Rancho Cucamonga, then report to the Dodgers.
Wilson is a three-time All-Star whose 48 saves in 2010 led the majors and tied a Giants single-season record. From 2008 through 2011, Wilson allowed only 15.1 percent of inherited runners to score -- the best rate in the majors.
Dodgers general manager Ned Colletti said the deal came together quickly Sunday after Los Angeles scouted the second of Wilson's two workouts at UCLA and witnessed him throwing 90-93 mph with what they considered good sink and a good cutter. The Dodgers had grown frustrated with the trade market for relievers, which Colletti called "overpriced."
Colletti said the Dodgers now are in the "listening stage" rather than actively seeking players before Wednesday afternoon's deadline.
"We know what kind of competitor [Wilson] is, we know what kind of games he's pitched in and the success he's had in the biggest games," Colletti said. "It's a matter, really of health and being sharp. And obviously we're going to take that chance.
"Most of the time when you're making a deal at this time of year, it costs you prospects and the cost of the contract. In this case, it's the cost of the contract."
Once he joins the Dodgers, Wilson likely will be used in a setup role. Colletti has been searching for another right-handed reliever to go with closer Kenley Jansen and setup men Ronald Belisario and Paco Rodriguez.
Since becoming the closer on June 11, Jansen is 2-0 with a 1.69 ERA and has saved 12 of his 14 opportunities. The Dodgers began the season with Brandon League as their closer before his struggles forced the change of plans.
Tuesday's move could mean the Dodgers are done dealing before Wednesday's trade deadline. Their primary focus was to add a reliever, though Colletti, who traded three minor-leaguers for Ricky Nolasco at the end of June, also has been having discussions about starting pitchers.
ESPN's Jim Bowden and ESPNLosAngeles.com's Mark Saxon contributed to this report.