Furyk, who now owns 11 top-10s this season, climbs to his highest career ranking.
Tiger Woods, who struggled to a 20th place finish at the U.S. Open, remained the No. 1 player in the world for a record 200th consecutive week. He has now been ranked No. 1 for 269 weeks during his career, trailing only Greg Norman, who held the top spot for 331 weeks during the 1980s and 1990s.
Ernie Els remained No. 2 behind Woods. Masters champion Mike Weir, who tied for third at the U.S. Open, climbed two spots to No. 3 in the world.
Davis Love III, who missed the cut, and Vijay Singh, who struggled to a final round eight-over 78 at the Open, both slipped one space to fourth and fifth, respectively.
Furyk jumped into the sixth position. David Toms moved up one to seventh, while Phil Mickelson slipped two spaces to No. 8. Padraig Harrington also dropped two places to ninth. Kenny Perry inched up one slot to 10th in the world.
Retief Goosen, the 2001 U.S. Open champion, dropped two notches to 11th. Nick Price, Justin Leonard, Chris DiMarco, Sergio Garcia and Robert Allenby remained in the 12th through 16th positions.
Bob Estes improved one space to No. 17, while Scott Hoch slipped one to 18th. Jerry Kelly held onto the 19th spot, while Scott Verplank moved up four places to round out the top 20. Rocco Mediate, who was 20th last week, fell to No. 25.
Leaney, who finished second in just his second U.S. Open, vaulted 32 spaces to No. 23 in the world.