Yes the One Ring or the Ring of Power controls the other rings.
Three Rings for the Elven-kings under the sky, Seven for the
Dwarf-lords in their halls of stone, Nine for Mortal Men doomed to
die, One for the Dark Lord on his dark throne In the Land of Mordor
where the Shadows lie. One Ring to rule them all, One Ring to find
them, One Ring to bring them all and in the darkness bind them In the
Land of Mordor where the Shadows lie."
—J.R.R. Tolkien's epigraph to The Lord of The Rings
However your mention of the three elvish rings is incorrect.
The greatest three Rings Celebrimbor crafted alone. So Sauron took no part in their crafting. He had no opportunity to corrupt them and turn them evil.
The Three that remained to the Elves were carefully hidden from
Sauron. They were not created as weapons of war or as a means to
dominate others; their purpose was to preserve the beautiful Elven
domains where their wielders resided, and to aid in healing and
resisting evil. However, they were still linked to the One Ring, and
whoever wore it could, with effort, see the thoughts of those who wore
So at most Sauron might have been able to read the thoughts of the elves (and Gandalf) wearing the three rings, but nothing more.
But he could not have turned them into wraiths, or controlled them.