He does not appear in novel "The Hobbit":
In Jackson's adaptation of the prequel, The Hobbit, Lee reprises his role as Saruman the White, even though Saruman does not appear in the novel. He is shown meeting with Gandalf, Galadriel and Elrond in Rivendell and speaking with them about the mysterious Necromancer encountered by Radagast the Brown, where he tells Gandalf that the Necromancer is just a mortal man and that Radagast is a foolish fellow because of his consumption of mushrooms.
Even though he is not in the novel, we could try and draw some conclusions from the novels. However, he has a different role there:
Saruman studied in depth the arts of Sauron in order to better oppose him. However, Saruman soon became enamored of Sauron's devices, especially the One Ring. He turned traitor to his mission, and sought power for himself by attempting to find the One Ring. He initially advocated an alliance with Sauron, but he soon betrayed Sauron as well, as his ultimate goal was to supplant Sauron and rule Middle-earth.
Source: LOTR Wikia
So, he never "falls under the thrall of Sauron" in the novels, but rather becomes the victim of his own greed for power.
My final answer to your question on his role in the movies is that we can only try to guess, and my guess would be "yes", not only for his somewhat obvious attempt to undermine Gandalf's warnings about the problem at hand, but also because it hardly makes sense for him to resist the power of Sauron for thousands of years, only to fall under it in the 50 or so years between the events of "The Hobbit" and "The Fellowship of the Ring".