Powell picked up 78 yards on 19 rushes while Greene donated 77 yards on 20 tries and each scored for the Jets (6-7), who won a second consecutive game for the first time this season.
Mark Sanchez came back from benching last week to connect on 12-of-19 throws for just 111 yards with a lost fumble for New York.
Ellis Lankster sealed the Jets' win with a late interception.
Chad Henne (21-of-43, 85 yards, two interceptions) struggled in Jacksonville's ninth loss in 10 games.
Montell Owens contributed 91 rushing yards on 14 carries for the Jaguars.
Celebrity Families of 2014 [PHOTOS]