Smith, 26, already leads the one-day side and will also captain the Twenty20 team in the absence of the injured Aaron Finch.
Fellow batsman David Warner, 28, has been named as Smith's vice-captain.
"At 26, Steve is a fine young man with extraordinary talent," said national selector Rod Marsh.
"He is highly regarded by the selectors and we congratulate him on being appointed to the role on an ongoing basis. He should be incredibly proud."
Clarke, 34, confirmed his decision to retire after his side surrendered the Ashes with a heavy defeat by England at Trent Bridge, which gave the hosts a 3-1 lead.
Smith had already captained the side in three Tests against India last summer, when Clarke was recovering from hamstring surgery and back issues.
The New South Welshman has played in 32 Tests for his country, scoring 2,952 runs at an average of 54.66.
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"When Michael made his decision to retire last week it was a very straightforward decision for us to nominate Steve as his successor," added Marsh.
"He has big shoes to fill but everything about him suggests he is the right man for the job."
Warner has earned the vice-captain's role despite a history of disciplinary problems, and Marsh said of the opener: "David has matured and developed into an important senior figure in the Australian team. He has come a long way.
"We believe that he will respond well to the added responsibility of leadership."
Smith and Warner will lead Australia in the limited-overs matches that follow the final Ashes Test at the Oval, which starts on 20 August.

Abdulmalik Adua, 17, of Woolwich and Rezwan Islam, 19, of Gloucester, were both sentenced to a minimum of 16 years in prison.
Paul Pass, 49, was stabbed in his flat in Gloucester last October by the pair, who used it as a drug-dealing base.
Adua and Islam were found guilty of murder at an earlier hearing. Islam was also found guilty of kidnap.
Bristol Crown Court heard Mr Pass was hit over the head with a guitar and stabbed in the leg, which caused his death.
Bilal Ebrahim Moosajee, 19, of Gloucester was jailed for nine years after admitting manslaughter and kidnap at an earlier hearing.
He will serve half his sentence before being considered for release on parole.
The court was told the men were all part of a drug-dealing gang from outside the county who were trying to establish themselves in the city.
They offered Mr Pass free drugs in exchange for using his flat as a base.
The gang took revenge by attacking Mr Pass after his friends had broken into the flat and stolen their drugs.
Mr Pass's mother Josephine said: "I feel both angry towards the men who took Paul from me and great sadness for their mothers.
"They have not only taken Paul's life, they have taken mine away as well because I feel like I have died."

Activists say about 35 pro-government fighters were killed and some medical staff taken hostage. More than 20 IS fighters were also reported killed.
The attack was centred on al-Assad hospital, to the west of the city.
Government forces retook the hospital after several hours, reports say, but the fate of the hostages is unclear.
BBC Arab Affairs Editor Sebastian Usher says the latest offensive shows that IS, despite recent setbacks, can still threaten significant government territory.
IS controls more than of half of Deir al-Zour and is seeking to capture the entire city.
The group has been besieging government-held areas there for two years, trapping about 200,000 civilians.
IS's Amaq news agency said its fighters had carried out a "major offensive" in Deir al-Zour early on Saturday, storming the hospital and cutting the route between a Syrian army base and the city's airport.
It said it had also taken over a fire station, university accommodation, grain silos and territory near the al-Tayyam oil fields.
A UK-based opposition news service, the Syrian Observatory for Human Rights, says government forces retook the hospital after several hours of fierce fighting.
Deir al-Zour is in an oil-rich area and on a vital supply route to the IS stronghold of Raqqa.
Steve Smith will become Australia's Test captain after the Ashes following Michael Clarke's decision to retire at the end of the series.
[NEXT_CONCEPT]
Two teenage drug dealers who murdered a man in a revenge attack have been jailed.
[NEXT_CONCEPT]
Militants from the Islamic State (IS) group have briefly overrun a hospital complex in the eastern Syrian city of Deir al-Zour, reports say.