The Mumbai Indians skipper will lead the Men in Blue in ODIs, T20s and will also serve as vice-captain in Test cricket.
Ashwin followed his six wickets in the first Test by picking up eight more in the second match against New Zealand.
With 124 rating points, India have replaced New Zealand in the position, who are on 121 points.
Laxman predicted that a three-figure score would come Virat Kohli's way soon, and that once it did, Kohli would be back to scoring big runs.
Chasing a mammoth target of 540, New Zealand were reduced to 140/5 at stumps on Day 3 after India declared their innings on 276/6 at Wankhede.
Mayank Agarwal scored 67 runs before being dismissed by Ajaz Patel, while Cheteshwar Pujara became the left-arm spinner's 12th wicket of the match.
New Zealand recorded their lowest ever Test score against India on the day as they were bowled out for just 62 runs.
Ajaz Patel created history as he took all 10 wickets in the Indian innings, making him the third bowler after Anil Kumble and Jim Laker to achieve the feat.
India started Day 2 of the second Test against New Zealand in Mumbai on 221 for 4 and finished with 285 for 6.
Ajaz Patel took his third five-wicket haul as India were rocked on day 2 of the Mumbai Test against New Zealand.