Mushfiqur, Sabbir, Mehedi and Mustafizur also move up.
Australia is now on 97 points, the same as New Zealand, but is behind on decimal points.
Lyon, Warner power series-saving Australia victory; Lyon takes best figures by an Australian in Asia
The visitors were 377 for nine, leading Bangladesh by 72 runs after David Warner smash back-to-back hundred.
The duo compiled an unbeaten 127-run stand for the third wicket to keep the hosts at bay. Aus were 225-2 at stumps on day 2.
Spinner takes five and Mushfiqur, Sabbir scores fighting fifties on hard-fought day one. Bangladesh were 253/6 at stumps.
Bangladesh eye historical series sweep against Australia the teams meet in the second and final Test, starting in Chittagong on Monday.
David Warner has returned to the top 10; career-best rankings for Tamim, Shakib; Stokes, Woakes and Cummins also move up.
David Warner century is vain, Bangladesh seal historic first Test win against Australia; Shakib Al Hasan picks ten wickets.
Vice-captain unbeaten on 72, Smith beside him on 25, and tourists needing a further 156 to win first Test against Bangladesh.