No-one ever really set the goalpost of the Space Race, so it can be whichever you want. I personally like to consider the Apollo–Soyuz mission as the true finis

No-one ever really set the goalpost of the Space Race, so it can be whichever you want. I personally like to consider the Apollo–Soyuz mission as the true finish of the space race, where the two nations docked their spacecraft together, and an astronaut and a cosmonaut shook hands in space. In the end the big winner of the Space Race was humanity, in the massive scientific leap forward that it created.