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 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.