In a Facebook post, Rajapaksa said: "As we usher in a new journey for Sri Lanka, we must remember that all Sri Lankans are part of this journey.
"Let us rejoice peacefully, with dignity and discipline in the same manner in which we campaigned," the Daily Mirror newspaper reported.
Gotabaya is retired soldier who took over Sri Lanka's defence portfolio during the period when his older brother, Mahinda Rajapaksa was President (2005-2015) and also when Sri Lanka ended its war in 2009 with the Liberation Tigers of Tamil Eelam (LTTE).
SLPP sources said that Gotabaya was likely to be sworn-in on Monday, reports the Sunday Times.
While congratulating Gotabaya on his victory, Premadasa in a statement said he would step down as Deputy Leader of the United National Party with immediate effect.
"At the conclusion of a hard fought and spirited election campaign, It is my privilege to honour the decision of the people and congratulate Gotabaya Rajapaksa on his election as the seventh President of Sri Lanka," the Daily Mirror quoted Premadasa as saying
"I express my heartfelt gratitude to all of our citizens who voted for me, in all corners of the island. I am humbled that you placed your faith in me. Your support has been a fountain of strength throughout my 26-year-long political career."
Official results were yet to be announced.
Sajith, the son of Ranasinghe Premadasa who served as the President from 1989 until he was assassinated in May 1993 by the LTTE, has pledged to fight for the Muslim and Tamil minorities.
According to preliminary results, Gotabaya secured 51.21 per cent votes, while Sajith managed to get 42.91 per cent of the ballots cast in Saturday's election.
Nearly 16 million Sri Lankans were eligible to cast their votes in the presidential election which was the third since the decades-old civil war came to an end in 2009, and the first after the April 21 Easter Sunday suicide bombings that took place seven months ago and killed over 250 people.
A record 35 candidates contested for the Presidency, but the main fight was confined between the two front-runners.
Saturday's percentage was a little less than the 2015 Presidential election when average voting percentage was 81.52 per cent out of which incumbent President Maithripala Sirisena won with 51.28 per cent with 6,217,162 votes while Mahinda Rajapaksa garnered 47.58 per cent with 5,768,090 votes.
( With inputs from IANS )