The Russian and European drafted resolution said that deployment of the UN observer mission, which will be called UNSMIS, will be "subject to assessment by the Secretary-General [Ban Ki-moon] of relevant developments on the ground, including the cessation of violence".
Saturday's resolution also noted that the cessation of violence by the government and opposition is "clearly incomplete".
Al Jazeera's Kristen Saloomey reports from the UN in New York.
The accusations have sparked fresh debate about the US mission in Afghanistan.
"What I saw bore no resemblance to rosy official statements by US military leaders about conditions on the ground," Colonel Davis wrote in an article published in Armed Forces Journal, a private newspaper not affiliated with the Pentagon.
"Instead, I witnessed the absence of success on virtually every level," he wrote under the headline, "Truth, Lies And Afghanistan: How military leaders have let us down".
"How many more men must die in support of a mission that is not succeeding ...?"
Troop commanders and politicians say the handover to Afghan troops is going well, but Lieutenant-Colonel Daniel Davis, who has just returned from the country, is challenging those statements in congress.
He says that conditions on the ground are ruinous - a conclusion deeply at odds with the picture of progress put forth by the top US military brass.
It is a rare instance of a US officer openly contradicting his superiors, and Davis' insights, first published in a military journal on Sunday, have been picked up by several members of congress.
At a Pentagon briefing on Wednesday, the US military's number-two commander in Afghanistan, Lieutenant General Curtis Scaparrotti, answered Davis' criticism, saying it was only one person's opinion of the general situation.
"I am confident, in my personal view, that our outlook is accurate," he said.
Scaparrotti says he does not doubt some of what Davis wrote, and he believes US forces have work to do in training Afghan forces.
Last week, Leon Panetta, the US defence secretary, said US forces would transition next year from a combat role to training Afghan soldiers and police.
