In the Anglo-American tradition, the shift was in the mid-1700s, in line both with the decline of the sword as a piece of daily wear, as well as the growing desire to place both duelists on as equal a footing as possible, easier done with pistol than sword.

In the Continental tradition, that is to say, principally France and Italy, the transition never really happened, and the pistol remained a rarity, with the majority of duels fought with swords right into the 20th century.

I have several responses which expand on this and can be found here.