To be completely honest, I don't think saying that one side "won" the Space Race is entirely correct. More to the point of the question: it did not "end" at Moon per se, but just slowly ramped down.
The support that NASA had in the initial stages of the space race was there because of the political reasoning - USSR put the first satellite in space, and this presented the US with a possibility of an orbital strike - nothing to laugh about. So more and more money was poured into NASA. Don't forget that the Space Race itself is very closely tied to the military in both countries. Space exploration was initially a secondary goal.
The Soviets developed their own rocket, the N-1, to go to the moon. However, that project was really not successful - early launch failures destroyed the N-1 launch pad, and after some more failures the project was eventually cancelled without ever having a successful test flight.
Don't forget that space is expensive. The Soviet Union was always hemorrhaging money and that only got worse during the years. On the US side, political support for Apollo died down after it had achieved the landing and funds were allocated for a more exciting project (from the perspective of the politicians), the STS. On the Soviet side, the effort was dedicated to building orbital station, which resulted in the Salyut (which was launched before Skylab) and Mir (the first modular space station!).
So, there wasn't really a meeting where it was decided that the space race was won. It's commonly considered that the US prevailed since putting men on the moon is really no easy task and requires a lot of new technologies to be developed. I personally think that the end of the Space Race is really the Souyz-Apollo joint mission, where the groundwork for the partnership between USA and USSR/Russia was laid. As with many things in history, the answer is not always clear-cut :)