Officers said they responded to the complex along Plaza Verde Drive just after midnight after receiving reports of a shooting and a suspicious person with a weapon.
Lt. Ali with HPD said once they arrived, officers found a man who had been shot in the chest. He said the man was later taken to the hospital, where he died.
The lieutenant added that it appears there was a disturbance inside one of the apartments, but homicide detectives are still investigating.
"We do not have a suspect in custody," Ali said. "(The suspect) fled on foot, and we don't have any information on him."
Police said they found a weapon at the scene but it's unclear if it belongs to the victim or the shooter.
Anyone with information on this case is urged to call HPD at 713-308-8800 or Crime Stoppers at 713-222-TIPS.