A spokesman for Murphy told People magazine that the "very intimate ceremony" was officiated by a rabbi and "they were surrounded by their families."
The newlyweds' first major public outing since getting married was at the Kentucky Derby last weekend.
Jeweler Neil Lane said the couple came to his store about two weeks ago to buy an engagement ring and when they couldn't settle on one, bought two. One was a five-carat platinum ring with a yellow diamond surrounded by white stones. The second was a six-and-a-half-carat platinum ring with a cushion cut diamond and diamonds embedded in the band.
"Simon told me: 'You know, you only live once,'" Lane said. "He did it with a lot of love and sentiment. I've never had that happen to me before. It's so romantic."
Murphy, 29, and Monjack, 37, will be working on a project together. Monjack will direct Murphy in "The White Hotel," which he adapted from D.M. Thomas' book about Sigmund Freud and an opera singer who was his patient.
2014: The Year in Music [PHOTOS]