MORE: Uncapped duo in Australia's Test squad
Ranked the world's best batsman in the shortest format of the game, Finch will lead Australia in a one-off T20 against Pakistan in the United Arab Emirates on October 5.
National selector Rod Marsh said Finch was "the obvious choice" after Bailey decided to step down.
"Aaron is the world's number-one ranked T20 international batsman and thoroughly deserves this opportunity," Marsh said in a statement.
"He has gained quality leadership experience through his role as captain of the Melbourne Renegades and the Pune Warriors.
"He also captained Australia A against the England Lions last year and excelled when given that chance.
"When George indicated that he was stepping down from the position, we thought Aaron was the obvious choice to succeed him.
"We know he will relish the opportunity to lead his country."
Included in the 13-man T20 squad were three uncapped players – all-rounder Sean Abbott, leg-spinner Cameron Boyce and paceman Kane Richardson.
Speedster Pat Cummins and Abbott are in the ODI squad who face Pakistan in three matches, while David Warner and Shane Watson have been named.
Phil Hughes, Richardson and Ben Cutting were dropped from the team which lost the tri-series final in Zimbabwe to South Africa.
Australia ODI squad: Michael Clarke (c), Sean Abbott, George Bailey, James Faulkner, Aaron Finch, Brad Haddin, Nathan Lyon, Mitchell Marsh, Glenn Maxwell, Mitchell Johnson, Steve Smith, Mitchell Starc, David Warner, Shane Watson.
Australia T20 squad: Aaron Finch (c), Sean Abbott, Cameron Boyce, Pat Cummins, James Faulkner, Brad Haddin, Mitchell Marsh, Glenn Maxwell, Kane Richardson, Steve Smith, Mitchell Starc, David Warner, Shane Watson.