Although the final tally was yet to be completed, Wickremesinghe's United National Party (UNP) has won 93 seats against the 83 won by United People's Freedom Alliance (UPFA) of Rajapaksa in a House of 225.
Election authorities said the UNP will fall short of an absolute majority of 113 by just four seats.
The 69-year-old two-time president was quoted as saying that he conceded defeat "after a good fight".
President Maithripala Sirisena, who defeated Rajapaksa in the January presidential polls, had earlier vowed not to make the former strongman premier even if his UPFA wins a majority.
"I offer my grateful thanks to all parties and individuals who worked untiringly during the election period to ensure victory for the people. We have been able to create a peaceful atmosphere needed for a free and fair election," the 66-year-old leader said.
He said people have backed the revolution achieved on January 8 and want that to continue.
"We have to unitedly work towards lifting the nation to a new height by facing challenges of the new era. I invite all of you to join hands with us to face the challenges of our mission," Wickremesinghe said.
In the north's Tamil districts, the Tamil National Alliance (TNA) appeared to have made a clean sweep by winning three districts.
A group of Sirisena supporters is likely to join a broad- based national unity government led by Wickremesinghe, who was appointed as Premier soon after Sirisena came to power.
Sirisena was Rajapaksa's health minister until he came forward as the opposition unity candidate to challenge the then president last year.
A total of 196 Parliament members are elected for a five-year term while 29 are appointed based on the national proportion of votes polled by each party.
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