The Soviets were the first to enter Vienna and took a majority of Eastern Austria if I remember correctly. So why was Austria allowed to be under Western influence?
Austria was not "given" to the West. Austria was an occupied nation much like Germany- divided up between American, French, British, and Soviet zones. Unlike Germany, there was a negotiated end to the military occupation in 1955 with the ratification of the Austrian State Treaty. The Austrian state pledged neutrality in European affairs, would not host military bases, and would never seek a reunion with Germany. The Soviets agreed to these conditions because it ended an occupation that was a drain on the Warsaw Pact's defenses and signaled a new turn in Soviet foreign policy after Stalin. Many in the US State Department saw the Austrian State Treaty as Austria undertaking a voluntary decision to become another Switzerland. There was little concern of "Finlandization" in which a peace treaty with the Soviets also carried certain defense obligations. The policies of subsequent Austrian governments bore out this as they hewed to a generally pro-Western foreign policy.
The Western influence of the Cold War was largely cultural. Austria's borders to the West were open, those to the Eastern bloc were, for the most part, not. Western products, tourists, and other connections integrated Austria into the West.