He told supporters at a ceremony at the presidential palace in Kinshasa he wanted to build a “reconciled Congo”.
Mr Tshisekedi was briefly taken ill before resuming his speech.
He is taking over from Joseph Kabila in the first peaceful transfer of power in the country in nearly 60 years, though many still dispute his victory in last month’s presidential election.
Kenya’s President Uhuru Kenyatta was the only African head of state, out of 17 invited, to watch the ceremony.
Numerous sources say his opponent Martin Fayulu won a landslide victory and has been denied office by a backroom deal between Mr Kabila and Mr Tshisekedi.
The influential Catholic Church, which deployed election observers on voting day, said the official data did not match its own numbers.
Mr Fayulu’s appeal to the Constitutional Court for a vote re-count was rejected, and critics say the body is too close to outgoing President Kabila and lacks independence.
Mr Tshisekedi had to cut short his inaugural address on Thursday after feeling faint. But he returned to the podium, saying he was exhausted from the long presidential campaign and the emotion of the day.
“We want to build a strong Congo, turned toward development in peace and security – a Congo for all in which everyone has a place,” he told the crowd.
Many people there believe the handover does show the DR Congo is becoming more democratic, the BBC’s Gaïus Kowene in Kinshasa reports.
Addressing the nation on Wednesday, Mr Kabila called on Congolese citizens to support his successor, who he said could call on him for advice at any time.
The new president is mostly known for being the son of the late veteran opposition leader Etienne Tshisekedi, who was regarded as one of DR Congo’s most important pro-democracy advocates.
The Union for Democracy and Social Progress (UDPS) party President Tshisekedi leads was founded in 1982 by his father, who turned it into the largest opposition party.
But President Tshisekedi, 55, insists he is not trying to rival his father.
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