Employees at McDonald's recognized him and delayed his food order in an effort to give police more time to catch up to Stephens.
Pennsylvania State Police say Steve Stephens was spotted Tuesday morning in Erie County, PA.
The chase lasted two miles before Stephens took his own life, police said.
Stephens kept posting on Facebook hours before his account was eventually shut down. Stephens had been employed as a case worker at a children's mental health facility.
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.
Stephens said in a separate video on Facebook on Sunday that he had already killed a dozen others.
The technology giant's chief executive said "our hearts go out to the friends and family" of Mr Godwin, who was shot dead by Steve Stephens in the USA city of Cleveland in Sunday in an apparent random killing.
Stephens had ordered a 20-piece McNuggets and a side of fries.
Lewis Hamilton apologises for pit-lane error in Bahrain Grand Prix
He was not happy with his McLaren car's performance, though, saying "I've never raced with less power in my life" over the radio. Mercedes have rules of engagement for its drivers but previously have always allowed them to race without orders.
The murder and a video sparked outrage across the world and renewed scrutiny of the growing number of grisly videos being posted on social media.
He then tried to buy some time for police by telling Stephens his fries "were going to be a minute".
The company has since announced it is launching a review for reporting harmful content.
Before murdering Godwin, Stephens had written on his Facebook page that he wanted to kill someone out of rage toward his ex-girlfriend.
Stephens was placed on the FBI's most wanted list and attracted a $50,000 bounty for information leading to his arrest.
After a lighthearted introduction in which the CEO of Facebook compared its F8 developer conference to the latest film in "The Fast and the Furious" franchise (which is also referred to as "F8"), Zuckerberg turned solemn.
Police said the woman he referred to was in a safe place.
"I don't want that man to die, I want him brought to justice", one of his sons, Robby Miller, said.