The situation unfolded in the 10100 block of Valley Club at about 10:30 p.m. on Saturday. Police said this started when a man arrived at the home to drop off children.
Lt. Riley of HPD said another man was inside the home, and the two men began arguing.
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The man in the home reportedly became upset and retrieved a handgun before opening fire at the second man. As the suspect began shooting, a woman also inside the home was injured in the gunfire, HPD said.
The man, who was shot, then fled the scene near Tidwell for help and was able to flag down officers in the area.
HPD said the suspect was arrested without incident.
Officials did not confirm who the woman was but said she was not involved in the initial incident.
Both victims were taken to the hospital, where they are stable.
The children inside the home were not harmed. Officials did not clarify the relationships between the two victims and the man who opened fire.