Rice admitted that a settlement would not be reached by the end of year but said: "This is the first time in almost a decade that Palestinians [and] Israelis are addressing all of the core issues to try to get to a comprehensive solution.
"It is really only possible to get to peace by dealing with all the core issues," she said.
The council had met in an emergency session on Saturday to receive the US and Russia-sponsored resolution.
Despite some lingering reservations from Libya and South Africa, the resolution was expected to be approved on Tuesday.
The text calls on Israelis and Palestinians "to fulfil their obligations" from talks begun at Annapolis, Maryland, last year, and for all nations and international bodies "to contribute to an atmosphere conducive to negotiations".
But analysts say the attempt to secure a resolution through the UN could be an attempt to secure the legacies of Rice and George Bush, the US president, after heavy criticism of their perceived inaction over the Israeli-Palestinian conflict.
Bush leaves office on January 20, when Barack Obama, the US president-elect, will take over as president.
The meeting was announced on Friday by Zalmay Khalilzad, the US ambassador to the UN, and Vitaly Churkin, his Russian counterpart, who both said the draft resolution was aimed at moving peace talks forward.
"The purpose would be to support the progress that has been made in the peace process and to encourage the sustainment and the successful conclusion of achieving the two-state solution and the Annapolis principles," Khalilzad said.
The talks have been stymied by ongoing violence, disputes over illegal Jewish settlement-building on Palestinian land and the future of Jerusalem, which both sides claim as a future capital.
Arab leaders also fear that the election of a right-wing, hawkish leader in Israel in elections in February could hamper the peace process further.