The 73-year-old took his oath at the tightly-guarded parliament complex on Thursday.
Mr Wickremesinghe - the former prime minister - is seen as deeply unpopular with the public, but some protesters have said they will give him a chance.
Sri Lanka has seen months of mass unrest over an economic crisis.
Many blame the Rajapaksa administration for mishandling the nation's finances, and see Mr Wickremesinghe as part of the problem. But there were few demonstrations on the streets the day after Mr Wickremesinghe won the parliament vote.
"He is here - and we will see what his actions are. If we don't get any food, any medicine, we'll be on the streets," one woman who had joined protests last week told the BBC.
Mr Wickremesinghe - who won a majority among lawmakers with the backing of the Rajapaksa's ruling party, the Sri Lanka's People's Front (SLPP) - took pains to distance himself from the old leaders.
"I am not a friend of the Rajapaksas. I am a friend of the people," he told reporters on Wednesday after defeating his main rival - a dissident SLPP MP - by 134 votes to 82.
On Thursday, the opposition signalled they would be willing to work with Mr Wickremesinghe. Local media is reporting that his cabinet is likely to include opposition MPs.
However, many protesters have expressed despair and disappointment over a Rajapaksa-allied politician's victory.
"I am absolutely disgusted at the result… I cannot believe that 134 people - MPs that are supposed to represent the people - have completely disregarded the wants of the people," activist Jeana De Zoysa told the BBC.
Soon after he was elected as president, Mr Wickremesinghe's message was clear.
He said any attempt to topple the government or occupy government buildings was not democracy and warned that those who indulge in such activities will be dealt with firmly according to the law. Mr Wickremesinghe said he was all for peaceful demonstration.
The protest movement now has to make tough choices whether to give some time for Mr Wickremesinghe to settle down and come out with his economic revival plan.
Many protesters were disappointed with his selection and wonder how a politician who could not defend his own parliamentary seat two years ago can become the country's leader.
Mr Wickremesinghe dismissed allegations that he had close ties with the Rajapaksas. But his critics question why the governing alliance of the Rajapaksas would nominate him and help win the presidency even though Mr Wickremesinghe was the lone MP from his own party.
Some are concerned that the government may gradually crackdown on the protest movement sooner or later.
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