The "Space Race" was not a thing with both sides agreeing on the finish line. It was a competition for space supremacy. The USSR had about 30 significant "firsts", while the US had about 33. That's pretty much a tie. More importantly, the USSR effort was mostly to build bigger stuff, while the US built better, more reliable, smarter stuff.
In terms of the cold war, the USSR's first ICBM (test) was launched in 21 Aug 1957, and the first strategic (nuclear) missile unit was operational in February 1959. The US ICBM (The Atlas D) was operational in October 1959. So this one goes to the Soviets, by eight months.
For a few years, the US official position that the Soviets had more nuclear missiles. But secretly, they knew that wasn't true. In 1961, for example, the secret count was USSR: 10 to 25, US: 57. By 1962, it wasn't a secret any more, that the US had several times more missiles, they were more reliable, and could be launched in minutes, compared to hours of preparation time for the USSR.
The "Prize" in the space race was peace, and both sides got that. I would say that the US got the better deal overall.