I'm speaking as a film and TV historian, so while I am familiar with 20th century US history in that capacity, I'm sure others will be able to offer more detailed politics on this issue.
The original Man From U.N.C.L.E. TV series was partly inspired by the resolution to the Bay of Pigs fiasco in 1962, when the US and USSR agreed to install a direct telephone hotline (that famous red telephone) between the White House and the Kremlin so that the two could have quick, direct interaction in times of crisis to avoid any such hostility, as well as the Partial Test Ban Treaty in 1963. While there was still much hostility between the two nations, these created a sense of hope that better diplomatic relations could be achieved, and opened an avenue for these sorts of fictional premises. Also keep in mind that The Man From U.N.C.L.E. was part of the "Bondmania" that arose in 1963 and '64, when everyone in film and TV wanted to jump on the James Bond superspy bandwagon (Ian Fleming was even hired to help develop the series' premise), and as such, The Man From U.N.C.L.E. promoted secretive international espionage and behind-the-scenes interactions between nations in order to stop supervillains even more dangerous than our Cold War adversaries.
While it came shortly after the Man From U.N.C.L.E. TV show and the setting of the film, the most relevant period of the Cold War to your question is the period of détente, a foreign policy that began in 1969 and lasted a little over a decade (until the Soviet invasion of Afghanistan and the Reagan presidency ended it, ramping up tensions again). Détente was a "thawing" of Cold War tensions between the US and USSR, characterized by a series of summits, treaties, and agreements between those two nations. This wasn't necessarily cooperation in the sense that you probably mean (Interpol-style supernation team-ups), but it was a move toward better relations between Cold War adversaries.
There was one remarkable bit of cooperation in 1960, when relations between the Americans and Soviets were very poor. Khrushchev and Eisenhower and many other world leaders were all in Manhattan for a UN session for an extended period, and all the leaders were meeting with each other, except Eisenhower and Khrushchev, who were not on speaking terms at that time. But they still managed to coordinate their schedules flawlessly, so that there was no chance that they would accidentally meet anywhere.