America was initially ignorant of the extent of the rift between the Soviet Union and China, but Richard Nixon and his Secretary of State Henry Kissinger managed to effectively split the communists when the opportunity arose.
China under Mao was far more similar to North Korea in regards to its isolation and unpredictability. On March 2, 1969, the Chinese attacked the Soviets on Zhenbao Island (Damansky Island), which is an island in the middle of the Ussuri River (a river forming the border between Manchuria and the Russian Far East). The Soviets suffered 59 dead and 94 wounded. The skirmishing went on until March 21, with the Chinese actually recovering a secret T-62 tank.
In August of 1969, the USSR sent its soldiers over the border (not a disputed island) in Xianjing, attacked and killed a Chinese squadron.
According to Henry Kissinger in his book On China, American intelligence literally did not know that China and the Soviets had entered into a small scale shooting war. The first big hint they had was when the Soviets urgently asked America how it would respond if it preemptively launched a nuclear preemptive attack against, what we though, was its ally China. Kissinger claims America assumed the Soviet Union was the aggressor (we had no diplomatic relationship with China), and the USA advised it would oppose such an attack. In this instance, it could be said America was discouraging a potential nuclear conflict.
By opening China, Nixon did the most any president did to "exploit" the rift. Neither China nor the USSR were particularly responsive to external manipulation. The modernization of China, in my opinion, should not be attributable to American influence. It is not so much as America "exploited" the rift, but pursued a policy of allowing American companies to do business in China.
However, the rift caused by Mao was definitely advantageous for America, but was more akin to foreign relations mana from heaven than anything earned or caused by America. In Kissinger's opinion, Mao did not think in western terms of juxtaposed states of War and Peace. He highlighted some Chinese historical practices of suddenly attacking your neighbors with incredible, overwhelming force, scarring the living shit out of them, and then voluntarily withdrawing back to the border before the other side can fully mobilize and is still reeling. I'm paraphrasing here, but basically the Chinese reasoning is similar to the boiling frog metaphor: Drop a frog in a boiling pot, and it will jump out and save its own life. Slowly bring the pot to a boil with the frog in it, and it will slowly drown and die. Put another way, if you slowly let relations deteriorate with an ally, tributary state, neighbor, etc., then they will mobilize for war and a long, drawn out slug-fest would result, or become harder for China to attack. If you attack them right at the start of a dispute, you force them to mentally confront how costly and unprofitable war with China is before they are invested in whatever the dispute is. Then, just as suddenly, they have all their territory back and China isn't asking you for anything, except to show some respect and quit acting like a bitch. It's hard for political leaders after such a shock, to counterattack China, but you know mentally they just punk'd you. This foreign policy mechanism worked for centuries against countries like Vietnam, India, and Korea. However, Mao couldn't treat the Soviet Union like that. Mao apparently didn't realize that something as minor as symbolically killing a few dozen Soviet border troops could threaten such a major alliance.
And China didn't stop using that technique after Mao, either. On November 23, 1978, Russia signed a 25-year mutual defense treaty with newly united Vietnam
On January 1, 1979, Deng Xiaoping told President Carter "This little child is getting naughty, it's time he be spanked."
Six weeks later, on February 15, the 1950 Sino-Soviet Treat of Friendship, Alliance and Mutual Assistance expired. That same day, Deng announced he was going to conduct a punitive attack on Vietnam, with whom the USSR had just made the equivalent of our own NATO allies. Deng also told the Soviets they had no quibbles about escalating to total war and left the bulk of the Chinese military on the Soviet border. Then, just as promised, they invaded North Vietnam, obliterated the troops between themselves and Hanoi, and left in March.
The Chinese cited some trumped up motives for supporting Cambodia and ethnic minorities in Vietnam, but the real reason was the mutual defense treaty with the Soviets. As humiliating as it was for the Soviets, they were not willing to go to war with millions of Chinese on their border over Vietnam. Instead, an alliance to deter invasion of Vietnam triggered its immediate invasion, and the Soviets did not intervene.
Nixon, in his book Leaders, noted that opening China unilaterally without consulting with the Soviets brought the People's Republic of China back into the U.N., and allowed it to counter the Soviets more effectively in international relations. By ending isolation of China, Nixon definitely eased the rift and confirmed PRC as an equal and counterweight to China. Otherwise, however, one must acknowledge that the rift between China and the Soviet Union is primarily due to the choices of Chinese leaders. If anything, Mao and Deng's actions were unnecessarily provocative and, regarding Vietnam, slightly humiliating.
In other words, America was not controlling or causing the split through its own agency. China was a juggernaut and far more willing to provoke the Soviet Union. However, America did not encourage the conflict between the two powers to escalate and, if anything, preferred to deter total war between the countries.
Sources:
Kissenger, Henry. On China. New York: Penguin Press. (2011).
Nixon, Richard. Leaders. New York: Warner Books. (1982).