Odoms played his entire NFL career with Denver. He was drafted fifth overall in the 1972 NFL Draft, which tied him with Mike Ditka for the highest drafted tight end ever in the NFL Draft. That record stood for 49 years until the 2021 NFL Draft, where Kyle Pitts was selected fourth overall.
Odoms was a four-time Pro Bowl selection and was a two-time All-Pro. He finished his career with 396 receptions for 5,755 yards and 41 touchdowns.