Beckett, 33, was 0-5 with a 5.19 ERA in eight starts and hasn't pitched since May 13. The Dodgers are hopeful he can pitch next season, when he'll be in the final year of a contract that will pay him $15.75 million.
The three-time All-Star joined the Dodgers last August in a blockbuster nine-player trade with the Red Sox along with first baseman Adrian Gonzalez, left fielder Carl Crawford and utilityman Nick Punto.
In the procedure performed in Dallas on Wednesday, Dr. Greg Pearl removed a rib to help treat a condition known as thoracic outlet syndrome. Pearl performed a similar procedure on St. Louis Cardinals pitcher Chris Carpenter last winter.
The Dodgers said Beckett will start his rehabilitation in five days and is expected to be ready to pitch by next spring.