The 35-year-old, who became the first English bowler, and only sixth ever, to reach 500 test wickets on Friday, moved above India's Ravindra Jadeja.
He is the oldest bowler to top the International Cricket Council (ICC) rankings since Muttiah Muralitharan in 2009.
Anderson took nine wickets in the series-deciding third test at Lord's which England won by nine wickets.
Ben Stokes, who produced a masterful spell of swing bowling to take six wickets in West Indies first innings before a vital 60 with the bat in a low-scoring game, replaced team mate Moeen Ali in fourth spot on the all-rounder's rankings.
Australia captain Steve Smith tops the batting rankings.
West Indies batsman Shai Hope, the top scorer in the series including two superb hundreds at Headingley, moved up 11 spots in the batting rankings to 30th.
(Reporting by Martyn Herman; Editing by Toby Davis)
