Pujara is the only Indian batsman inside the top-20 of the rankings. Among the bowlers, off-spinner R Ashwin dropped a rung to be eighth in the latest list, while Zaheer Khan is placed 17th.
Elsewhere in the list, England players made big gains in the rankings, a day after Alastair Cook's side won the second Ashes Test against Australia at Lord's by 347 runs.
Bell, who won the ICC Emerging Cricketer of the Year Award in 2006, is now 11th in the rankings for Batsmen, just three ratings points behind New Zealand's Ross Taylor after jumping six spots.
A right-handed batsman, Bell made scores of 109 and 74 in the Lord's Test.
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England opener Root, who won the Player of the Match award for his 468-minute knock of 180, has moved up 21 places to 26th. He is now the sixth England batsman inside the top 30.
The batting table is still headed by South Africa's Hashim Amla with West Indies' Shivnarine Chanderpaul in second place and AB de Villiers in third position.
Meanwhile, in the rankings for Bowlers, James Anderson moved up one place to join Peter Siddle in fifth spot, and both the bowlers now trail fourth-placed Saeed Ajmal of Pakistan by 25 ratings points.
England off-spinner Swann, who ended with a match haul of nine for 122 in the Lord's Test, has gained one place and is now seventh, just five ratings points behind Anderson and Siddle.