Ashwin hit an excellent 106 in India's second innings and took eight wickets with his off-spin in the second Test against England in Chennai.
Kohli, who had had scores of 11 and 72 in the opening Test against England, has 852 points and down one spot in the latest chart issued on Wednesday.
Apart from Kohli (862) and Pujara (760), Test vice-captain Ajinkya Rahane (748) is the other Indian batsman in the top-10, hanging on to the eighth spot.
Paceman Kyle Jamieson continued an extraordinary start to his test career, taking 6-48 as New Zealand beat Pakistan by an innings and 176 runs in the second test.
New Zealand will have to win the series against Pakistan and expect a tie in the Border-Gavaskar series to retain their No.1 ranking.
The win takes New Zealand to the top of the ICC Test Rankings, for the first time since the inception of the concept, with 117 ratings.
Although he remained at the second spot in the ICC rankings, the half-century helped him gain two points to reduce the gap between him and Smith from 25 to 13
In the list of all-rounders, Ravindra Jadeja is the highest-ranked Indian, currently occupying the third spot. Ashwin is placed at the sixth spot.
The Black Caps defeat the visitors by an innings and 12 runs early on the fourth day of the second Test to grab the first spot in rankings for the first time in history.
James Anderson has climbed to eighth spot in the latest ICC Test Rankings, released after England's series against Pakistan ended on Tuesday.
Broad has moved up one slot to the second position in the bowlers' list after a haul of 4/56 while Anderson has moved up two slots to 14th with figures of 3/60, as the pair helped dismiss Pakistan for 236 in their only innings of the match.
Gambhir claims that Virat Kohli and Co. have played more competitive cricket away from home than other teams as they won a Test match in South Africa, England and also won the Test series down under in Australia.
After a tumultuous transition phase post the ball-tampering scandal, Australia on Friday displaced India as number one side in Test format but Langer is aware that it does not take long for situation to change.
Following the update done by ICC on Friday, India's results in the 2016/17 season, where they had won 12 Tests and lost just one, were eliminated while also reducing the weightage of games played until May 2019.
Virat Kohli dropped to No.2 with 906 points to his name after a poor show with the bat in the first New Zealand Test at Basin Reserve.
Indian captain Virat Kohli remains at the top of the ICC Test Rankings, which were updated on Friday.
Virat Kohli stays at the top of the rankings, and Marnus Labuschagne jumped a place to reach 3rd in the latest Test rankings.
With 928 points, Kohli is way ahead of second-placed Australian maestro Steve Smith (911) and New Zealand's inspirational skipper Kane Williamson (822).
10 years hence, India have two more ICC trophies to their name - both in the 50-over format and both under Dhoni, Kohli has transformed into a run machine, all certain to break Tendulkar's once-invincible record, while the nation has emerged as an almighty force in Test cricket with a prized collection of formidable pacers who can dominate even on home turfs.
Vaughan also praised Australia and felt their position in the ranking charts doesn't match as they are currently placed at the fifth spot.
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