Let's face it, both sides were rushing to be first pretty much regardless.
The Soviet rocket, which was designated N1 was kept extremely secret, primarily it was being developed by the Progress and Nikolai-Kolmykov bureaus in Samara - I've met a few of the Engineers who worked on the design and even now they are cagey about it. I've never seen drawings or models, but it was described to me as "very much like the Saturn V". Hardly surprising, since the state of the art in modern rocket technology was fairly similar both sides of the iron curtain.
A problem of the era is that unlike in the US, the Engineers couldn't readily walk away from political pressure when it all got too difficult - and that lack of ability certainly led to a few too-many corners being cut occasionally - I suspect that Tu-144 and Buran were particularly bad examples of this.
However, when the US landed on the moon with Apollo 11, the decision from the Kremlin was to simply bin it all and pretend that the project never happened. When you consider the amount of brilliant work that must have been buried, a real tragedy - what could have happened if they'd carried on?
G