The White House official said Mr Burns and Mr Naryshkin, the head of Russia's SVR spy agency, did not discuss any settlement of the war during their meeting in Ankara.
A Kremlin spokesman confirmed to Russia's state news agency that talks between the two did take place.
The meeting was the highest-ranking direct engagement between a US and Russian official since before Moscow's invasion of Ukraine on February 24.
Ukrainian officials were briefed before Mr Burns's travel to Turkey.
He also planned to discuss the cases of detained US citizens Brittney Griner and Paul Whelan, the release of whom US President Joe Biden's administration has been seeking.
The US reportedly has offered convicted Russian arms dealer Viktor Bout in exchange for Ms Griner and Mr Whelan. Russia has yet to respond to the offer.
The meeting between the spy chiefs came as Mr Biden met with Chinese President Xi Jinping on the sidelines of the G20 summit in Indonesia.
Mr Biden and Mr Xi noted their “opposition to the use or threat of use of nuclear weapons in Ukraine”, the White House said.
The leaders also agreed that nuclear war “should never be fought and can never be won”.
Mr Biden last month warned that the risk of a nuclear “Armageddon” was at its highest level since the Cuban Missile Crisis in 1962.
Washington has warned for months that Russia could use nuclear force as Moscow continues to face setbacks in its invasion.
Agencies contributed to this report