The new President will have a mandate to serve out the rest of Rajapaksa's term, which ends in November 2024.
Prime Minister Wickremesinghe was sworn-in before Chief Justice Jayantha Jayasuriya, Daily Mirror reported.
The Parliament would be summoned on Saturday (July 16) to take nominations for the next President and then to have an election among 225 MPs.
India has time and again proved to be a true friend of Sri Lankan people in their need, the minister said.
He also referred to India's financial aid to help the island country deal with its severe economic crisis.
President Rajapaksa was not present in the presidential House when a mob stormed the building on Saturday.
Angered by the attack, the protesters had set fire to the house of Wickremesinghe who had left it with his wife.
The 53-year-old Jayasuriya said he always stands with the people of Sri Lanka adding that the people will celebrate victory soon.
So as to ensure the safety of the citizens, the prime minister said he is agreeable to this recommendation by the Opposition party leaders.
According to sources, the President had been moved out of his residence on Friday, in anticipation of Saturday's protests.
According to the Prime Minister's Office, the premier also has requested the Speaker to summon Parliament in an effort to find a solution to the crisis.
The event saw participation of the Governor of the Central Bank of Sri Lanka Nandalal Weerasinghe, senior officials, representatives from various businesses and industries.
A total of close to 400,000 MT of various types of fuel have been delivered with Indian assistance in March and April.
The hike came as the public continues to suffer in long queues at fuel stations hit by shortages.
India on Saturday provided another 40,000 metric tonnes of diesel to Sri Lanka under the credit line facility.
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