The team made the announcement shortly before Tuesday night's game against the New York Mets at Citi Field.
The Giants said they will make a corresponding roster move Wednesday.
Melancon, 32, signed a four-year, $62 million contract as a free agent in December. The right-hander has a 2.53 ERA in 11 appearances and has converted six of his first eight save opportunities for the Giants, who are last in the National League West with an 11-22 record.
Derek Law, the only other Giants reliever with a save this season, appears to be the top candidate to close games for manager Bruce Bochy in Melancon's absence. For now, Bochy said, the team will go with a closer-by-committee approach.
Melancon said the elbow has been bothering him on and off since the latter part of spring training. The Giants sent him for an MRI during their weekend series inCincinnati. When it revealed some inflammation, the team placed Melancon on the DL retroactive to May 6.
"We want to get this cleared up so it's not nagging him all year," Bochy said. "We think he should be ready when his time [on the DL] is up."
Melancon will be eligible to come off the DL next Tuesday.
Melancon has played nine seasons in the majors and made 455 career relief appearances. He has never been on the disabled list.
"It's really tough,'' he said. "I've never been on the DL, and it's been a huge goal of mine to do that my whole career. But I expect to be back as soon as possible. I feel like this team is going to turn it around. And once we do that, I want to make sure that all cylinders are clicking."
Information from The Associated Press was used in this report.