TEHRAN, Young Journalists Club (YJC) - The Central Election Commission said Jeenbekov had won around 55 percent of the vote, with oligarch opponent Omurbek Babanov taking more than a third of the ballot.
"Jeenbekov is leading," CEC head Nurzhan Shayldabekova told reporters in the capital Bishkek, saying 95 percent of the ballots had been counted.
Preliminary figures showed turnout at just over 50 percent.
"Our campaign took great pleasure in the favorable results announced by the CEC," said Konduz Joldubayev, a spokesman for Jeenbekov.
However, a spokeswoman for Babanov told AFP they were conducting "a parallel count".
Pre-election campaigning was overshadowed by an apparent smear campaign against Babanov, who leads the parliament's second largest party, and suspicions that Jeenbekov could benefit from government patronage.
Jeenbekov is a close ally of incumbent president Atambayev, 61, who chose to respect a single, six-year presidential term and step down.
Resource-poor Kyrgyzstan is seen as the most democratic state in Central Asia, a predominantly authoritarian region, but it has also been the most politically volatile in recent times.
The Muslim-majority country experienced two revolutions that unseated presidents in 2005 and 2010 followed by ethnic violence that left over 400 people dead.
Sunday's election featuring 11 candidates promised to see the first peaceful transfer of power between two elected presidents but the build up to the vote was beset by tensions.