In addition to my comment on the question itself, I have an answer that dates to the beginning of the modern international system (or just before, if we take the Peace of Westphalia as our starting point).

During the 30 Years War, England supported the cause of the Elector Palatine against the Holy Roman Emperor in a phase of the conflict that centered on the invitation of the Elector to take the throne of Bohemia, over the protests of the Hapsburg monarchy. During the second phase of the war, France and Sweden both intervened under the pretext of defending "German liberties," inserting themselves into Imperial politics (and advancing their territorial claims) by military means.

The famous Spanish Armada offers an even earlier example; to use the language of the Cold War heyday for American interventions, the Catholic Spanish wanted to impose a client regime over England, thereby replacing the ideologically rival Anglican-Protestant monarchy. In less anachronistic terms, the Spanish saw Elizabeth I as an illegitimate usurper of Queen Mary I of England (also the wife of Phillip II of Spain). Ironically, the invasion was not only ideologically motivated (and, in turn, endorsed morally and economically by the Papacy... another case of intervention) but strategical impelled by English aid to the rebellious Spanish Netherlands.

As I write this, I realize that one could go much further by following the thread of Papal interventions throughout European history and, for that matter, interventions against the Papal State and/or the Catholic hierarchy (often to secure the election of a preferred candidate to the Holy See). A few examples...

Papal condemnation of the Cathar Heresy and theological endorsement of a Frankish invasion of Languedoc during the Albigensian Crusade.

Papal economic support for the restoration of the Medici as masters of Florence.

Papal negotiations with Francis I of France and Charles V of the HRE over the suzerainty of Naples (i.e. negotiating the foreign policy status of an outside polity with two third parties)

Sources: Peter H. Wilson, "The Thirty Years War: Europe's Tragedy"

Lauro Martines, "Furies: War in Europe, 1450-1700"

R.J. Knecht, "Francis I"

Strayer, "The Albigensian Crusade"

TL:DR If you'll pardon this Early Modernist's digressions, you'll find that intervention in the politics of other polities (countries, if you prefer) is neither a recent development nor unique to the US. Other commenters might offer insight into even earlier cases.

I see the famous "Zimmerman Telegram" as fitting the description. In 1917, German foreign secretary Arthur Zimmerman sent a secret correspondence to Mexico offering German support for Mexico for reconquering “the lost territory in Texas, New Mexico, and Arizona” in exchange for a Mexican attack on the United States. It certainly has shades of the Bay of Pigs in Cuba... The Telegram also pushes Mexico to ally with Japan and coordinate their attack with a Japanese invasion of the West Coast.

The Zimmerman Telegram: America Enters the War, 1917-1918

Another possible candidate may be the various instances that Britain and France have instigated Native American tribes to attack America -- both American Colonies (French and Indian War) and the United States of America (War of 1812). Although the United States started the latter war, Britain had been cultivating Indian tribes for years in advance for the purpose of making war on the Americans

In 1976, the House Intelligence Committee found that the Greek junta known as 'The Colonels' had contributed $500k to the 1968 election campaign of Nixon & Spiro Agnew. They also gave funds to the 1972 campaign.

Prior to receiving this money, Agnew (who was of Greek descent) had voiced strict neutrality about the unelected junta. Agnew then reversed himself and lauded the regime two days prior to a Greek referendum. The junta press carried Agnew's endorsement as front-page news, and they won their referendum.

Agnew visited Greece in 1971, where he again applauded the junta as the best rulers of Greece since antiquity. When Congress banned aid to the junta, the White House exploited a national security loophole to evade the restrictions and continue aid.

The junta's liaison with Washington, Thomas Pappas, also claimed to have obtained a $25k donation for Nixon's CREEP (Committee to Re-Elect the President), in exchange for a fuel contract for the 6th Fleet for a Greek businessman. (who was awarded the contract despite not being the lowest bidder).

Complicating all of this was the fact that the money was siphoned from Greece's intelligence service, which was in turn subsidized by the CIA, so an argument could be made that it was the CIA's covert action funds that were bankrolling US presidential candidates.

Agnew of course was forced to resign when confronted with charges of tax evasion and accepting bribes during his tenure as governor of Maryland.

Are there any examples from post-WWII?