Can you be more specific about what Carlin says? That episode is quite long, so I'm not sure what was said or how detailed it was.

Christopher Andrew's "Sword and Shield", based on Soviet sources, does say that Khruschev asked the Washington KGB resident to "propose diplomatic or propaganda initiatives, or any other measures to facilitate Kennedy's victory" out of fear that Nixon was hawkish enough to start a nuclear war. Apparently the KGB tried to contact Robert Kennedy, but were rebuffed.

It'd be characteristic of Service A, the KGB's arm devoted to active measures as opposed to intelligence gathering, to overstate it's influence and claim they had significantly influenced the election. Most of Service A's efforts to plant rumors and conspiracy theories had little success, but even a few clippings in obscure newspapers and magazines were evaluated as highly successful in influencing popular opinion. I don't have any info of any specific actions taken during the 1960 election by the KGB but they do seem to have at least considered trying.

Nikita Khrushchev indeed claimed that Kennedy was elected because of him. In his memoirs (Memoirs of Nikita Khrushchev: Volume 3: Statesman, 1953–1964, 293-296) he gave the following account of the United States presidential election of 1960:

Although Khrushchev claimed that all presidential candidates were the same to the Soviet Union because they all were representatives of the big capitalists, he admitted that there were "shades of differences" between the candidates, and because of this the Soviet Union preferred one over the other.

Khrushchev's favorites for president were either Eisenhower or Adlai Stevenson, but Eisenhower was not allowed to run for a third term and Stevenson failed to be nominated by the Democratic Party because he had already lost two presidential elections and so Kennedy was nominated, because he was young and a millionaire.

Between Kennedy and Vice President Nixon, the Republican candidate, the Soviet Union preferred Kennedy because they knew almost nothing about him, while Nixon was known in the Soviet Union for his "aggressiveness toward the USSR, the anti-Communism that he preached, and his former ties with the reactionary and obscurantist Senator Joseph McCarthy".

On May 1, 1960, the American pilot Francis Gary Powers was shot down during a spy mission in the Soviet Union, and two other American pilots were shot down on July 1, 1960, over the Barents Sea. According to Khrushchev, at "the culminating stage of the election campaign, just before the voting", the USA requested an amnesty for Powers and other prisoners. While the Soviet Union was generally in favor of this, Krushchev said to the members of the Politburo:

“We shouldn’t do this now, because the presidential candidates are trying to use this issue to their advantage, and voices can be heard on the radio saying that this or that candidate is better able to ensure good relations between the USA and the USSR.” (In fact they said “with Khrushchev personally.” The capitalist press always tries to single out a specific individual rather than emphasize the social position of this or that person, that is, the position they hold in their society.) “If we release the prisoners now, that will be to Nixon’s advantage. Even the slightest shift that might tip the scales in his favor would not be good for us. Let’s not take this step now, because I don’t expect that our relations will improve if Nixon becomes president.”

So, Krushchev feared a release of Powers and other Americans in Soviet captivity at this time would be seen in the American public as a political success for the Republicans. This would improve the chances of Nixon to become president.

Kennedy won, but only by a slim margin. Krushchev claimed this victory was only possible because he didn't release the prisoners before the election. When he met Kennedy in 1961, in Vienna, he told him:

At one point I asked him: “Mr. Kennedy, do you know that we voted for you?”

He looked at me quizzically: “How so? How is that to be understood?”

I informed him about the appeal from Washington to Moscow just before the end of the election campaign, giving the exact date, and said that if we had returned Powers and the others at that point, it would have been considered an accomplishment of Nixon’s.

He began to laugh and when he had collected himself he replied: “You’ve drawn the right conclusion. I agree that, just at that time, even the slightest shift in the balance could have been decisive. So I grant your point that you took part in the elections and voted in my favor.”

This joking reflected reality.

So yes, Krushtchev claimed that Kennedy was elected with the help of the Soviet Union. But is this true?

Memoirs are difficult sources because they are notoriously untrustworthy. Most politicians who write memoirs want to explain themselves and tend to overstate their achievements. Political bias and false memories are other problems. So, memoirs tend to have factual errors, oversimplification and wrong interpretation of events. Krushchev is no exception. One can see this in this account. The nomination process of the Democratic Party in 1960, for example, is oversimplified: There were more than two candidates, and Stevenson was not even the runner-up (he won just 5,25 % of the votes on the Democratic National Convention).

It is also dubious that Krushshev wanted to grant Powers amnesty. Gary Powers was a Soviet prisoner until he was exchanged for a Soviet spy on February 1962, more than a year after Kennedy became president. But it is true that Krushchev released the two American pilots shot down over the Barents Sea as a goodwill gesture after Kennedy was inaugurated.

After a quick research I couldn't find any sources which confirm that the US government requested an amnesty for Powers days before the election. Perhaps some user here with a better knowledge about the US foreign policy during the Cold War can help.

But even if this is true, it is difficult to say that Kennedy won the election because the Soviet Union didn't release Gary Powers. At most this contributed to Kennedy's victory, but it is hardly the only reason.