What are some major military successes achieved by the former Soviet Union against the Western World that shook the US and its Cold War allies?
There were none. Likewise, the U.S. never scored any military victories over the Soviet Union.
The Cold War was a war by proxy. One superpower was a combatant in Afghanistan, Korea, and Vietnam, but the other did not send troops to the other side — partly out of fear of escalation into global conflict. Rather, they provided support to their allied local factions, as both did in the numerous other conflicts of the Cold War.
That is not to say that other military conflicts did not shake the Western powers. The single most significant development was the defeat of the Nationalists in the Chinese Civil War and the declaration of the People's Republic of China in 1949. The shock and the ensuing bitter debate over who had "lost" China colored American policy for a generation.
The answer to your question depends heavily on how you define "Military". As Choster's answer implies, defining military successes as direct military action between the armed forces of different nations means that the Soviet Union and the US never triumphed military over each other. Indeed, except for some aerial skirmishes during the Korean War, Soviet and American forces did not ever face off against each other for an extended period of time.
That said, direct Soviet military aide through advisers and equipment, as well as diplomatic pressure, was vital to defeating or at least defending against direct and proxy US military action all over the world, but especially in East Asia. Rather you define that as military action or not is hard to say, and I am not expert enough to qualify how critical soviet military aide and diplomatic pressure was in the American defeat in Vietnam for instance, but it is fair to say that America's defeat in Vietnam had a profound effect on the entire Western world, and that Soviet support undoubtedly played a role.