I don't really see it as a crucial turning point. Perhaps if it had ended differently it might have been a turning point in the other direction, in Russia's favor. But a "turning point" implies that something changes in an important way, and the Cold War continued for several decades after that.
Maybe our pullout in Vietnam was more of a turning point. At that point we were no longer facing a Russian ally in war for the first time in a long time.
I believe it was a turning point because the perception was that the Russians backed down, and Khrushchev was seen as weaker than Kennedy. That perception would haunt Khruschev and to many it was the beginning of the decline of the old guard Soviet dominance. For more information on the Cuban Missile Crisis check out the link below.
I think the Cuban Missile Crisis made everyone, on both sides of the Cold War, really stop and consider the implications of the capabilities of nuclear weapons. For the first time since the United States bombs fell on Japan to end World War II (which was a completely different set of circumstances and was not something the public was aware of ahead of time), the general population was faced with considering the impact of bombs falling upon their homes and the retaliatory bombs that would be launched in response.
I think the Crisis, while it may not have brought about an immediate change, did plant the seeds of realization that there had to be another way to deal with differences than through threat of military force.
I suppose you could say that it was in the sense that it made the US and the USSR more aware of how dangerous the Cold War was. You could argue that it dissuaded them from ever getting close to actually fighting one another again. However, I don't really think it was a turning point in allowing the US to win the Cold War.
So when did the general population consider the impact of bombs falling upon their homes and the retaliatory bombs that would be launched in response?
My grandfather says On the Beach, the 1959 movie which predated the Cuban Missile Crisis, focused his attention in that direction.
What might be the turning point of the Cold War?
Maybe the Berlin Airlift because it was the first time the west stood up to Stalin.
Maybe the Soviet invasion of Afghanistan because it drained an already fragile Soviet economy.
Maybe the death of Leonid Brezhnev because of the weak leaders who followed him.