I agree that the success of the Mongols under Genghis Khan and the success of Japan in modernizing and becoming a major power in the Meiji era would be considered wildly improbable if they had not happened.
One of my favorite -
Spain's extraordinary success in the Americas. Both Cortez and Pizarro invaded huge powers under especially favorable circumstances. The Aztec Emperor Montezuma was superstitious and too cautious and essentially invited Cortez' Army into Tenochtitlan instead of fighting them. The Tlaxcalans attacked Cortez's force and were wearing them down by attrition when they decided to ally with Cortez against the Aztecs instead of fighting him. The Spanish governor of Cuba sent a large expedition to arrest Cortez and his supporters for exceeding their authority, but Cortez was able to induce part of this army to desert to his side and defeated the rest in a surprise night attack. One member of this second expedition had smallpox, which spread to local Indians and then spread like wildfire through the Aztec Empire, weakening their ability to resist. Cortez' Tlaxcalan allies remained loyal to him even though they also suffered heavily from the smallpox. Even with all of this, Cortez and several of his senior captains were almost captured during the final assault on Tenochtitlan.
With Pizarro, he arrived after smallpox had already swept through the Inca Empire after spreading from Spanish settlements on the north coast of South America. This plague killed many, including the experienced old Emperor and his favorite son. Two of his other sons began a civil war for the throne. Atahuallpa, one of the sons, was so fixated on fighting his brother that he did not take the Spanish threat seriously. He and many of his commanders and nobles met the Spaniards unarmed and assuming that the Spanish would not dare attack them. Instead, the Spanish attacked and captured Atahuallpa and killed or captured most of his leading nobles. Holding Atahuallpa as a hostage, Pizarro defeated the other side in the civil war under the guise of supporting Atahuallpa, then accused Atahuallpa of plotting against him and had him executed. By the time the remaining Inca forces started to coordinate their fight against the Spanish, the Spanish were intrenched in several key parts of the former Inca empire, including the capital of Cuzco.
One of these two stories would seem lucky but believable, but both of them happening within 15 years was extraordinary good luck on the part of the Spanish and gave them access to huge amounts of land and wealth in the Americas.