Now in defense of the US, the soviets did put a satellite up first, but we were more than capable of doing so at the same time or even before. We sent up a satellite 2 months after the soviets. Contrast that with the moon landing. The soviets never sent a man to the moon even decades after we did. So in that light, I think the moon landing was far more impressive and representative of "victory" for a lack of a better word.
Perhaps we can say the soviets won the first space battle, but we won the space war? Without a doubt, we won the PR war as the moon landing and armstrong's "That's one small step for man, one giant leap for mankind" will be the iconic and defining moment of the space race for the rest of history.