"The Greek people gave us a strong mandate to change Greece. We will honour it to the full," the 51-year-old Harvard graduate and former McKinsey consultant said after taking his oath of office in a televised ceremony, his wife and three children close by.
"Hard work begins today. I am completely confident that we will prove equal to the challenge."
Mitsotakis later named a cabinet with many technocrats in junior posts and only a handful of women.
It will be sworn in on Tuesday and meet on Wednesday, he said.
The new finance minister is Christos Staikouras, a 45-year-old former junior finance minister who holds a banking PhD from London's City University.
The foreign ministry goes to Nikos Dendias, 59, who was public order minister when a historic probe began into the neo-Nazi Golden Dawn party following the 2013 murder of an anti-fascist rapper.
Among notable choices is the appointment of two former far-right party allies to head development and agriculture.
The US-educated conservative faces a hefty challenge as he takes over from Tsipras, who imposed austerity measures required under a bailout to keep Greece in the eurozone.
The country's public debt last year stood at 335 billion euros ($376 billion), or 180 percent of GDP.
The debt load is forecast to fall to 168 percent of GDP this year, but only throught belt-tightening brought in under Tsipras's Syriza party, policies which Mitsotakis's New Democracy party says is stifling growth.
The tricky job of keeping Greece's international creditors onside while easing the hardship on Greeks -- by lowering taxes and renegotiating fiscal targets -- could result in a short honeymoon phase for Mitsotakis.
An upcoming meeting of eurozone finance ministers will discuss the state of Greece's economy after tax cuts rolled out by Tsipras in a last-ditch attempt to win over voters.
- 'Message for change' -
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Former banker Mitsotakis has pledged to create jobs and "steamroll" obstacles to business.
Those pledges wooed Greek voters, who handed him nearly 40 percent of the vote in Sunday's election, well ahead of the 31.5 percent given to Tsipras.
"It's a strong message for change in Greece," Mitsotakis told reporters on Sunday.
Last week he told AFP his mission was "to make sure we restart the economy" with "ambitious growth driven by private investments, exports and innovation".
He predicted he could persuade Greece's creditors to accept the easing of tight fiscal targets with "a comprehensive reforms package".
Tsipras, for his part, warned that Mitsotakis would do away with the social spending he brought in to help Greece's vulnerable groups.
He portrayed the Mitsotakis family -- one of Greece's leading political dynasties -- as part of a failed system that bankrupted the country in 2010.
- Son of a PM -
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Mitsotakis is the son of former prime minister Constantine Mitsotakis, one of the country's longest-serving parliamentarians. His sister Dora Bakoyannis is a former minister and Athens's first female mayor.
And in a first even for a country accustomed to family links in politics, the recently-elected mayor of Athens is the incoming prime minister's nephew.
Mitsotakis took charge of New Democracy three years ago. Following the election, his party will have a majority of 158 seats in the 300-member parliament.
New Democracy was last in power in 2014, in coalition with the Greek socialists.
Syriza's parliamentary presence will shrink from 144 seats to 86. Tsipras has promised to remain an "active" voice in opposition in defence of labour rights.
Sunday's election was Greece's third in as many months.
In May, New Democracy beat Syriza by nearly 9.5 points in European parliament elections. A week later, it completed a near-sweep of Greek regions in local elections. After that, Tsipras was forced to call an early general vote.
Entering parliament for the first time are Greek Solution, a nationalist party formed by TV salesman Kyriakos Velopoulos, and MeRA25, an anti-austerity party founded by maverick economist and Tsipras' former finance minister Yanis Varoufakis.
Neo-Nazi party Golden Dawn was shut out of parliament for the first time since 2012.
Golden Dawn, until recently Greece's third-ranking party, is in steep decline amid an ongoing trial for the 2013 murder of an anti-fascist rapper, allegedly carried out with the knowledge of senior Golden Dawn members.