The Grizzlies announced the deal Monday.
In the trade, the Grizzlies send forward Tayshaun Prince to Boston and forward Quincy Pondexter to New Orleans. The Grizzlies also acquired point guard Russ Smith, a 2014 second-round pick, from New Orleans and a trade exception. The Pelicans send Austin Rivers to Boston in the deal.
Memphis also sends a 2015 second-round pick to New Orleans and a protected future first-round pick to Boston.
Green is expected to provide more offense for Memphis, which is tied with Houston atop the tight Southwest Division.
Boston Celtics president of basketball operations Danny Ainge said that the uncertainty surrounding Jeff Green's future with the team, along with the value he commanded on the trade market, ultimately led to the three-team swap.
"Any time you lose your leading scorer, there's going to be challenges that come with that," Ainge said. "Losing him three weeks after losing your multiple-time All-Star point guard, there's going to be challenges that come with that."
Rivers, the son of former Celtics coach Doc Rivers, was the 10th overall pick in the 2012 draft. Sources say that Boston does not intend to keep Rivers and will make him immediately available via trade. One source with knowledge of the Clippers' thinking told ESPN.com's Ramona Shelburne that Los Angeles -- with Doc Rivers serving as coach and team president -- believes it will ultimately acquire the coach's son.
The Riverses have said in the past they would be against being on the same team, but when Doc was asked Saturday if he would be willing to coach Austin if the opportunity should present itself, he sounded more open to the idea than ever before.
"I would," Doc Rivers said. "I think a year ago I probably wouldn't. I think I would for sure. I think this team could handle that. He's a downhill guard, which is something we need, so I certainly would [be open to coaching him]."
ESPN.com reported earlier this week that the Grizzlies were aggressively pursuing Green or Miami Heat swingman Luol Deng in hopes of bolstering their wing rotation in the ever-competitive Western Conference. Green became the focus, sources said, when it became apparent that Miami was not ready to entertain offers for Deng, who is not even halfway through the first year of a two-year, $20 million contract.
ESPN.com's Marc Stein, Arash Markazi, ESPNBoston.com's Chris Forsberg and The Associated Press contributed to this report.