Multiple sources said that Addis Ababa had dropped its proposal on the understanding that no further renewal of the probe would be sought by Western countries when it expires later this year.
"Western countries sought to stop them (filing the motion) by coming to an understanding, as there was a risk Ethiopia could have won that vote," said Marc Limon of Universal Rights Group think tank.
Ethiopia had circulated a draft motion to be submitted to the current rights council session in Geneva seeking to end the inquiry six months early in a never-before-seen move that would also block the publication of its findings and a final debate.
Ethiopia has opposed the investigation from the outset, calling it politically-motivated and trying to block its funding, preferring national accountability efforts.
But five diplomats and human rights sources said Ethiopia had since backed off amid pressure.
Ethiopia's diplomatic mission in Geneva did not respond to Reuters questions.
The probe has been losing political support amid a broader push-back against what is perceived as a Western-dominated human rights agenda, and many African states oppose it.
The deadline for submitting resolutions was Thursday at 12:00 GMT and the Ethiopia resolution was not on the list, a U.N. website showed. The 47-member Geneva body does not have legal powers but sometimes its probes lead to cases before national and international courts.
The International Commission on Ethiopia, the only independent probe into the two-year conflict which pitted Ethiopia's army against forces in the northern Tigray region, had already found reasonable grounds to believe that all parties have committed war crimes.
It is now also investigating "serious violations" committed since a November peace deal. The U.S. also determined this week that all sides including the Ethiopian and Eritrean armies had committed war crimes - allegations they both reject.