The swap of minor leaguers was announced on Wednesday.
Elliott and Lindberg will report to the American Hockey League's Belleville Senators. Sexton and Erkamps have been assigned to the AHL's Wilkes-Barre/Scranton Penguins.
The 27-year-old Elliott has eight goals and 16 assists in 84 NHL games with Colorado, Arizona and Nashville. He last appeared in the NHL in April 2016 with the Predators.