Ashwin’s match haul of 10-225, that helped notch a 197-run victory in the first Test in Kanpur on Monday, saw him overtake England’s James Anderson by one point.
He is now within seven points of number-one ranked Dale Steyn of South Africa, who leads the rankings for Test Bowlers with 878 points.
Another strong performance in the Kolkata Test will potentially see Ashwin top the bowling charts for the second time in 2016 after first claiming the number-one position last year following the Boxing Day Tests.
Williamson similarly edged England’s Joe Root by one point with innings of 75 and 25 to reach 879 points in the rankings for Test batsmen, which is led by Australia captain Steve Smith with 906 points.
However, to move ahead or edge even closer to Smith, Williamson will have to produce strong performances in the remaining two Tests.
Ashwin also contributed with a useful innings of 40 to consolidate his position at the top of the list of all-rounders. Ashwin reached a career-best rating of 450, opening up a 66-point gap with second-placed Shakib Al Hasan of Bangladesh.
There was not much to cheer for other New Zealand batsmen apart from Williamson, but for India, opener Murali Vijay and Cheteshwar Pujara made significant gains.
Vijay and Pujara, who became the first Indian pair in 10 years to notch century partnerships in both innings of a Test match, moved up from joint-20th position to joint-16th position.
Other Indian batsmen to move up were Lokesh Rahul, who gained five ranks to reach 57th position, and Rohit Sharma, who gained two slots to reach 52nd.
Victory in their 500th Test means India is now one victory away from reclaiming the ICC Test Championship mace from Pakistan. India had entered the series trailing Pakistan by one point and needed to win the series to move ahead of its traditional rival.