This Senses of Cinema article quotes art historian Marilyn Stokstad's definition of a term for the dramatic use of light and shadow known as tenebrism as, "The use of chiaroscuro and artificially illuminated areas to create a dramatic contrast of light and dark in a painting.” Caravaggio's emphasis on light and use of chirascuro are evident in the Orson Welles film, Citizen Kane. The article describes how Caravaggio's work is referenced early in the film, just after the opening scenes.
Also referencing Citizen Kane, Lighting as Storytelling from Cinematography Theory and Practice by Blain Brown explains further how light is used to tell the story in the same ways as Caravaggio uses light in his paintings.
Carvaggio's influence can be seen in Mel Gibson's The Passion of the Christ in both Caleb Deschanel's cinematography and Maurizio Millenotti's costume designs. According to this article about the production, much of the film was shot in the dark to achieve the desired interplay of light and shadow. Caravaggio's sense of scale as well as his preference to portray the grittier side of realism is evident, as is the sense that the action is barely contained within the frames of the image, enhanced with the use of strong foreshortening of form.
The Passion of the Christ Trailer
Another film that looks to Caravaggio is The Black Dahlia. The American Society of Cinematographers Magazine quotes cinematographer Vilmos Zsigmond's approach of looking to strong, real light sources to create his effect, saying that, "When you’re doing a crime film, you have to create shadows. The Black Dahlia was certainly that kind of movie, so I couldn’t think of a more appropriate way to light it.”
A lesser known example, although striking in the resemblance to Caravaggio's work is this Greek film, Black Field.
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