England won the match by 141 runs to take a 2-1 lead over Pakistan.
Anderson had entered the Edgbaston Test trailing Ashwin by one point, but he has now opened up a 12-point gap over the India spinner.
This is following Anderson's four wickets against third-ranked Pakistan, which has earned him six points, as compared to Ashwin who has dropped six points despite taking six wickets against eighth-ranked West Indies in the Jamaica Test.
But the competition at the top of the table is so tight that the number-one ranking has thrice changed hands from Yasir Shah to Ashwin and back to Anderson.
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Australia's Mitchell Starc, Sri Lanka's Dilruwan Perera and Chris Woakes of England are the other bowlers to make impressive gains in the latest rankings.
Perera has vaulted 11 places to a career-best 16th position following his 10 wickets in Galle, making him the third Sri Lanka bowler inside the top 20 after Rangana Herath (eighth, up by one place) and Dhammika Prasad (19th, down by one place).
In the Edgbaston Test, he took five wickets, which has helped him rise three places. Woakes now has 39 wickets in 11 Tests, including 31 wickets in five home Tests this season.
Batsmen to make upward movement following the Edgbaston, Galle and Jamaica Tests include Alastair Cook to seventh (up by one place), David Warner to ninth (up by one place), Angelo Mathews to 10th (up by three places), Ajinkya Rahane to 11th (up by four places), Jonny Bairstow to 15th (up by one place), Kusal Mendis to 17th (up by four places), Azhar Ali to 18th (up by eight places), BJ Watling and Tom Latham to joint-28th, Moeen Ali to 33rd (up by eight places), Gary Ballance to 36th (up by five places) and Jermaine Blackwood 44th (up by 10 places).
In the ICC Player Rankings for Test all-rounders, Moeen Ali has achieved a career-high third position following scores of 63 and 86 not out as well as match figures of two for 128. Ashwin is the number-one ranked all-rounder, while Ali's England team-mate Stuart Broad is fifth after slipping two places.
The player rankings will be now be updated following the Colombo Test.