Family of the victims said it was the son's 16th birthday.
The crash happened at about 8:40 p.m.
Police say the officer was responding to reports of an ATM being robbed, when he T-boned the victims' car as they were pulling out of a Dollar Tree parking lot.
Witnesses described the scene, saying they heard a loud sound, then saw the police cruiser crush the other vehicle and catch fire.
The victims' identities have not been released.
Hours after the crash happened, a person was found in the back of the police cruiser, authorities told ABC Houston affiliate KTRK said.
Video shows first responders frantically pulling him out of the cruiser to give him aid.
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However, police haven't been able to explain why that person was in the back of the patrol car or why it took so long to discover the person was there.
"That really got to me. Three and a half hours later they just now find him? That means they didn't check if anyone was in the car," said witness Kenneth Mitchell, who told KTRK he was at the Dollar Tree when the crash happened.
The conditions of the officer and the person in the back of the police cruiser weren't immediately known.
KTRK is also working to learn what happened to the suspected ATM robber.
Missouri City police said the officer has been with the department just under a year. He came to Missouri City from a prior agency, where he had about two years of experience.