Deputies responded to an assault call in the 5600 block of Wigton around 10:30 a.m.
When authorities arrived in the area, they found the suspect with a knife in his hand just a block away in the 10300 block of Mullins, according to Harris County Constable Ted Heap.
Heap said the suspect attacked the deputy, and in a reaction of self-defense, the deputy discharged his firearm and struck the suspect.
The suspect was taken to a local hospital in stable condition.
The two people injured were also taken to the hospital for apparent stab wounds, authorities said. They, too, were in stable condition.
The Houston Police Department arrived to the scene to conduct an investigation.