Haley said in a statement that the opening "will allow UN peacekeepers to step up their efforts to prevent hostilities in the Golan Heights region."
The United Nations Disengagement Observer Force (UNDOF) has been monitoring operations in a demilitarized zone established in 1974 between the Golan Heights and the Syrian sector, but the peacekeeping mission was disrupted by Syria's civil war.
Israel captured the Golan Heights in the 1967 Middle East war and fought Syria again on the strategic plateau in a 1973 conflict.
"We look to both Israel and Syria to provide UN peacekeepers the access they need as well as assurances of their safety. We also call on Syria to take the necessary steps so UNDOF can safely and effectively deploy and patrol without interference," Haley said.
Military police from Russia, a major ally of Syrian President Bashar al-Assad, have been patrolling on the Syrian side of Quneitra.
Israeli Defence Minister Avigdor Lieberman visited the Israeli side of the crossing on September 27. He said then that in the past, traffic through Quneitra mainly comprised shipments into Syria of apples grown by Druze farmers on the Israeli-controlled Golan - and the entry of brides for partners on both sides of the armistice line.
Want to enjoy 'Zen' reading - with no ads and just the article? Subscribe todaySubscribe now