It ended a few miles away when police bumped Stephens' vehicle to get it to stop.
Williams says investigators had hoped to talk with Stephens to find out a motive.
Steve Stephens was wanted in a slaying that he apparently recorded Sunday in Cleveland, and posted on Facebook, setting off a manhunt that spread Monday across the north eastern United States.
On Sunday, Stephens uploaded three videos to Facebook.
Authorities in Cleveland urged a nationwide debate on the issue of violence on social media and to find answers to the plague of gun crime rocking America.
The case prompted Facebook to review how quickly and easily its users can report material that violates standards. "I snapped. Dog, I just snapped, dog".
One of Godwin's daughters, Debbie Godwin, said she wished Stephens had been captured.
"I'm not happy he's dead at all, not at all". Cleveland police previously warned that Stephens could have escaped anywhere between the states of Pennsylvania, New York, Michigan, and Indiana.
Early Tuesday, someone called police to report that he thought he had seen Stephens at a hotel in Washington, D.C., but a police spokesman there said authorities quickly determined that the person was not the man being sought.
When officers confronted Stephens, he fled in his vehicle before later being stopped, the chief said, adding: "As the officers approached that vehicle, Steve Stephens took his own life".
"I forgive him because we are all sinners", Robby Miller, Godwin's son, said in an interview with CNN. The Ford Fusion did a half-turn and came to rest at the curb. However, Godwin-Baines saw the footage and became emotional as she recalled seeing the fear in her father's eyes moments before he lost his life. The one officer just shook his head.
Police said it wasn't clear whether Stephens had any help while he was on the run or where he had been and that investigators will try to retrace his steps.
Law enforcement officials said his cellphone was last tracked Sunday afternoon in Erie, Pennsylvania, about 100 miles (160 kilometers) east of Cleveland.
"I found somebody I'm about to kill", Stephens said in the video.
"This is something that should not have been shared around the world, period", said Williams, the police chief.
Stevens was spotted by a worker at a McDonald's drive-thru near Erie, Pa., police said.
In the video of the shooting, Stephens told Godwin the name of his girlfriend and said, "She's the reason that this is about to happen to you". Godwin did not seem to recognize the name.
The woman Stephens spoke of, Joy Lane, said in a text message to CBS that "we had been in a relationship for several years".
In the video, Stephens claimed to have killed more than a dozen people, police said, although they have not confirmed other victims.