Selectors also axed wicketkeeper Matthew Wade from the 15-man squad for the tournament in India. Test gloveman Peter Nevill, who has never played a limited overs international, was picked instead.
Opener Finch, who has been struggling with a hamstring injury, was named captain in September 2014, but only skippered the side on six occasions as Australia concentrated on other forms of the game.
Finch remains in the squad released on Tuesday for the tournament in India.
Smith's elevation makes him captain in all three formats of the game.
“We think now is the right time for Steve to lead Australia in all three forms of the game as it offers us important continuity, not only ahead of the World T20, but beyond that tournament as well,” selectors chief Rod Marsh said in a media release.
“(Finch) will have benefited enormously from the leadership opportunity and will remain a highly-respected leader within the Australian squad.”
Wade, who had been Australia's limited-overs wicketkeeper since Brad Haddin's retirement from the short formats following last year's World Cup success, paid the price for his recent patchy form behind the stumps.
Nevill had a modest output with the bat during Australia's domestic 'Big Bash' T20 tournament, but Marsh said the squad had enough batting depth to pick him on his glove work alone.
“We want Australia's best wicket-keeper playing in this tournament and we consider Peter Nevill to be the best in the country right now,” he said.
Left-arm spinner Ashton Agar, who has never played any T20 cricket, was also named in the squad along with another potential debutant in legspinner Adam Zampa.
The pair complement the off-spin of all-rounder Glenn Maxwell, with test spinner Nathan Lyon overlooked.
Finch, fast bowler Nathan Coulter-Nile (shoulder) and all-rounder James Faulkner (hamstring) have all been named subject to fitness.
Batsman Usman Khawaja has been named in the squad after a prolific home summer in tests against New Zealand and West Indies and in the Big Bash tournament.
Australia have never won the World T20 title in five previous attempts, their best finish coming in West Indies in 2010 when they lost to England in the final.
Squad: Steve Smith (captain), David Warner, Ashton Agar, Nathan Coulter-Nile, James Faulkner, Aaron Finch, John Hastings, Josh Hazlewood, Usman Khawaja, Mitchell Marsh, Glenn Maxwell, Peter Nevill, Andrew Tye, Shane Watson, Adam Zampa.
Original source: Smith helms Australia for World T20