Vijay has moved up three places to 20th, while Dhawan has vaulted 15 places to 45th. Ajinkya Rahane is the other notable mover and despite missing out on a century, he has climbed up four places to a career-high 22nd to date, the ICC said in a statement.
However, Kohli has slipped a rung to 11th.
In the bowlers' chart, Indian off-spinner Ravichandran Ashwin and Bangladesh's Shakib Al Hasan have gained one place each and are now 12th and 16th, respectively.
Despite missing out on what would have been his first double century in his 28th Test, Smith has leapfrogged Kumar Sangakkara of Sri Lanka and South Africa duo of AB de Villiers and Hashim Amla following scores of 199 and 54 not out in the latest rankings.
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However, Sangakkara will have a chance to reclaim his number-one ranking when he features in the series against Pakistan, which starts in Galle on Wednesday.
In achieving the landmark, Smith has also become the 23rd Australia batsman to date to top the batting table, with the last being captain Michael Clarke in January 2013.
Steve Waugh spent 94 Tests as number-one -- more than any other Australia batsman -- while Don Bradman spent a record 6,320 days as the top-ranked batsman -- more than anyone else in the history of the game.
In the Rankings for Test bowlers, Australia fast bowlers Mitchell Starc and Josh Hazlewood have also achieved career-best rankings to date.
Starc has moved up four places to 21st after match figures of four for 84, while Hazlewood has also gained four places following match figures of seven for 56 and is now in 27th spot.