The treaty, which was signed in 1991, committed both parties to limit arms stockpiles, including a maximum of 1,600 missiles and 6,000 warheads.
Friday's meeting, meanwhile, marked the highest-profile talks between the two nations since Barack Obama, the US president, took office in January.
The Geneva dinner was intended to set the stage for next month's meeting between Obama and his Russian counterpart, Dmitry Medvedev.
Clinton, the US's senior-most diplomat, called the talks a "fresh start" in relations between the two after tensions rose last year over Russia's brief war with neighbouring Georgia, a US ally.
"I think we can arrive at a common view, both in the context of strategic offensive weapon and missile defence"
Sergei Lavrov, Russia's foreign minister
Another issue of contention has been US plans to set up a missile defence shield in Eastern Europe, which Russia considers its back yard.
Sounding an optimistic note, Lavrov told the joint news conference: "I think we can arrive at a common view, both in the context of strategic offensive weapon and missile defence."
In a light-hearted gesture, Clinton handed Lavrov a red button reading "reset" in English at the start of the meeting.
However, the Russian translation of the word read "peregruzka" (overload), instead of "perezagruzka" (reset).
Lavrov joked later that they had "reached an agreement on how resetting should sound in both Russian and English".
On a more serious note, the Russian foreign minister expressed hopes of strengthening US-Russian economic ties.
"We have a common interest in a new level of economic relationship between the two countries," he said.
Clinton said the discussions also touched on the two nations' mutual interest in advancing nuclear disarmament and on growing concerns about Iranian nuclear development and instability in Afghanistan.