Jones, a retired US general, told Russian Foreign Minister Sergei Lavrov that President Barack Obama remained committed to mending ties with Moscow after they were badly strained in recent years.
"I would like to... on behalf of President Obama, reassure you and your colleagues that the path that US and Russian relations are on right now is one that's full of promise and potential," Jones said.
"We want to do everything we can to bring that good state of an affairs to a conclusion," Jones added, at a meeting also included the top US and Russian negotiators working on the Strategic Arms Reduction Treaty (START).
Washington and Moscow have been seeking to agree a replacement for START, a landmark 1991 treaty that led to steep cuts in the US and Russian nuclear arsenals, before it expires on December 5.
"We believe that only the most intense work will allow us to fulfill the agreement by our presidents to do everything possible to reach a new agreement on strategic arms reductions before the current treaty expires," Lavrov said.
Besides START, the Iranian nuclear programme was also expected to be on the agenda at Thursday's talks, Russia's ITAR-TASS state news agency reported ahead of the meeting, citing a foreign ministry source.
Obama has vowed to "reset" US-Russian relations that hit a post-Cold War low during the previous administration of George W. Bush, due to disputes over issues such as missile defence and last year's Russia-Georgia war.
But Moscow and Washington still have differences on key matters such as how the international community should respond to Iran's controversial nuclear programme.