It was a case of spies watching spies watching spies: Israeli intelligence officers looked on in real time as Russian government hackers searched computers around the world for the code names of American intelligence programs.

What gave the Russian hacking, detected more than two years ago, such global reach was its improvised search tool — antivirus software made by a Russian company, Kaspersky Lab, that is used by 400 million people worldwide, including by officials at some two dozen American government agencies.

The Israeli officials who had hacked into Kaspersky’s own network alerted the United States to the broad Russian intrusion, which has not been previously reported, leading to a decision just last month to order Kaspersky software removed from government computers.