It happened around 2:00 near the Citibus station.
Lubbock County Sheriff Kelly Rowe said a few of his undercover officers were working in the area when a person took off running from the investigators.
The plain-clothed officer chased after the suspect, but fell during that pursuit and broke his arm.
Sheriff Rowe said undercover officers, like the one hurt Monday, are working around the clock to make Lubbock County a safer place to live.
"There is a tremendous amount of narcotics activity happening in Lubbock and at all different levels and we're constantly trying to track that down and find that and get it off the street," Rowe said.
Another officer was able to catch the suspect and take him into custody.
Sheriff Rowe said that person was wanted on an outstanding warrant.