Tillerson's statement sought to tie the need for Russian cooperation in Syria to the US mission to defeat ISIS, saying, "In order to complete the mission, the global community, and especially Russia, must remove obstacles to the defeat of ISIS and help provide stability that prevents ISIS from rising anew from the ashes of their failed and fraudulent caliphate".
The US launched a cruise missile attack on a Syrian air base in April in response to what it said was a regime chemical weapons attack on civilians.
The issue is expected to be discussed during the first meeting between Russian President Vladimir Putin and his USA counterpart Donald Trump at the G20 summit this week in northern German city of Hamburg.
Russian Federation and the USA had succeeded in establishing "de-confliction zones" to prevent their forces attacking one another, Tillerson said, which showed "that our two nations are capable of further progress".
"The USA and Russian Federation must publicly commit to protecting civilians in Syria and to ending violations by their own forces as well as by the warring parties on the ground".
Bashar al-Jaafari, Syria's ambassador to the United Nations and leader of the Syrian government delegation to Astana, attributed the failure of the meeting to reach a final agreement on "counter-efforts" of the Turkish side, reports SANA.
During his daily call with reporters, Kremlin spokesman Dmitry Peskov declined to comment on Tillerson's remarks.
Jaberi Ansari also held talks with the Kazakh foreign minister, Kairat Abdrakhmanov, during which the two sides exchanged views on the latest developments regarding the Astana conference and the current efforts for resolving the crisis in the war-torn country.
USA -backed forces have surrounded Daesh stronghold in Syria, the city of Raqqa.
The remarks offer the latest stop on a bumpy US policy ride that has left worldwide observers with a case of diplomatic whiplash as they try to figure out whether the Trump administration will insist that Assad step down from power.
At the G20 in Hamburg this Friday, face-to-face talks are expected betweeen US President Donald Trump and Russian President Vladimir Putin.