At least 2,000 Tshisekedi supporters gather outside the Palace of the Nation, the seat of the presidency, to celebrate the historic event.
Tshisekedi was declared winner with 38.5 percent of the vote, over his opposition rival Martin Fayulu, who was credited with 34.8 percent.
At 55, opposition leader Tshisekedi takes over from Kabila, 47, who is stepping aside after 18 years at the helm.
Opposition leader Felix Tshisekedi was sworn in Thursday as president of Democratic Republic of Congo, in the country's first peaceful handover of power since it gained independence in 1960.
Tshisekedi, 55, took the oath of office before receiving the national flag and a copy of the constitution from outgoing president Joseph Kabila, who is stepping aside after 18 years at the helm of sub-Saharan Africa's biggest country.
He was briefly taken ill during his inaugural address but returned to the stage to complete his remarks.
Thousands of Tshisekedi supporters, many of them dressed in white, celebrated the historic event outside the Palace of the Nation, the seat of the presidency.
"We hope that this will be a real change, especially as he has taken power without bloodshed," said Saddam Kongolo, a member of Tshisekedi's Union for Democracy and Social Progress (UDPS).
Task
One of Tshisekedi's first tasks will be to appoint a prime minister in a move which will see him sharing power with Kabila's supporters, who hold an overwhelming majority in parliament.
The ceremony caps more than two years of turmoil sparked by Kabila's refusal to step down when he reached the constitutional limit on his term in office.
A country the size of continental western Europe, DR Congo has lived through two regional wars in 1996-97 and 1998-2003.
The last two presidential elections, in 2006 and 2011 — both of which were won by Kabila — were marred by bloody clashes.
The ballot, which took place on December 30 after being delayed three times, surprised many by the lack of violence, but a political storm swiftly brewed over the count.
Opposition
Tshisekedi was declared winner with 38.5 percent of the vote, over his opposition rival Martin Fayulu, who was credited with 34.8 percent.
Fayulu branded the result a fix but lost a challenge to the Constitutional Court and foreign support for his position ebbed as countries took comfort in a peaceful transition.
Among foreign nations attending the ceremony, Kenya and Zambia were represented by their presidents and Tanzania by its vice presidents, according to the RTNC state television, while China, France, Japan and the United States sent their ambassadors, AFP journalists saw.
"The opposition has run out of recourse to challenge the election results and the threat of widespread post-election violence is gradually subsiding," said Robert Besseling of EXX Africa, a business risk consultancy.
The UDPS is DR Congo's oldest and largest opposition party, which was founded and led by his father Etienne.
Analysts say he faces a raft of pressing problems after taking the oath of office.
Challenges
He must defuse the anger of Fayulu's supporters, carry out his campaign pledge of ending the "gangrene" of corruption after the Kabila era, and forge a power-sharing arrangement with the outgoing president's bloc.
The pro-Kabila Joint Front for Congo (FCC) controls 337 seats in the 500-member National Assembly against 102 for Fayulu's coalition, Lamuka, and 46 for his own coalition, Heading for Change (Cach).
"Tshisekedi will have little margin for manoeuvre," said Stephanie Wolters at South Africa's Institute for Security Studies (ISS) think tank.
According to a "political coalition agreement" seen by AFP which outlines an arrangement for "power-sharing" between the FCC and Cach, the position of prime minister "will be rotated every five years."
The defence, foreign affairs and interior portfolios will go to "the president's political family".
Kabila himself will become a senator for life, a position reserved for outgoing presidents under the constitution. He is widely expected to wield influence through his supporters.
His family has acquired a wide range of assets during the many years in power.
Beyond the politics, Tsishekedi also has to deal with the brutal militias who control parts of the country's strife-torn east where an Ebola epidemic is also unfolding.
He also has to meet the expectations of his supporters about easing poverty, which afflicts the vast majority of the country's 80 million citizens.
Despite the poverty, DR Congo boasts a treasure trove of minerals, ranging from gold and diamonds to copper and coltan — a mineral essential for the batteries used in hand-held electronic devices.
Yet, in a country where graft is entrenched, very little of the wealth trickles down to the poor. It ranks a mere 176th on the 189-nation Human Development Index compiled by the United Nations' Development Programme (UNDP).
