Barney later died at the hospital, the sheriff added.
Shortly thereafter, about 10 a.m., officers encountered a male later identified as 31-year-old Cory Lee Henderson walking near 2100 E. 4500 South, Winder said.
"As soon as officers came into contact with this individual, gunfire erupted," Winder said. "Multiple rounds were exchanged both from the suspect and from our officers."
The same neighbor who saw officers helping Barney described the shots as "a hail of gunfire," and estimated there were about 20-25 rounds fired within a 5-second time frame.
Henderson died during the shootout and a second officer, Jon Richey, 51, was hit, Winder said. A single shot went through both of his legs, Bell added.
As of Sunday evening, Richey had improved from serious to fair condition and was "doing remarkably well," Bell said.
Police are still looking for the female who was allegedly with Henderson in the car at the time of the crash, and who they describe as a person of interest. Salt Lake City police will be investigating the shooting and is asking the public to call them with any information at 801-799-3000.
Barney was a father of three children who had battled cancer twice, according to West Jordan Detective Brent Jex, who is the president of Utah's Fraternal Order of Police. In 2010, Eisenhower Junior High School held a fundraiser to help Barney with his cancer fight, the Granite School District added.
"Officer Barney represents the best of the profession and his positive energy was contagious to all around him. The profession is worse off without him," said Jex, who knew Barney well. "Doug Barney embodied what the nobility of policing is all about."
Barney also worked extra security shifts at the KUTV news station, according to one of its anchors and reporters, Amy Nay. He would walk with the staff to their cars and was there Saturday night with "his big smile and presence," Nay tweeted. "'Stay safe,' he said."
Gov. Gary Herbert ordered the U.S. and Utah flags be lowered on the day of Barney's funeral.
"Today a husband and father left his wife and three children, put on a badge, and promised to return home soon," Herbert said in a statement. "On behalf of Utah residents from across the state, our hearts are broken for the family of Officer Doug Barney who gave the ultimate sacrifice today while in the line of duty." Herbert and his wife, Jeanette, also extended their prayers to Richey and his family.
"Every day, we send our men and women in uniform into harm's way so that we may live in peace and safety," Herbert said. "On a day like today, we are reminded that we will never be able to thank them and their families enough for the sacrifices they have made."
Salt Lake City Police Chief Mike Brown also offered his condolences to both Barney and Richey's families.
"It doesn't matter what badge you wear, or what name is on your badge, we all in law enforcement lost a great officer today and so did the community in which they serve," Brown said.
While police were still at the scene of the shooting, Henderson's mother and brother arrived and were "extremely distraught," Winder said.
"These scenes are extremely difficult for everybody involved," Winder said."... Unfortunately, force needed to be used to restrain [the mother]."
The brother got out and reportedly told officers to leave his mother alone, but was arrested after allegedly punching an officer after an officer had grabbed his wrist to control him, according to a jail document. He also had outstanding warrants and was booked into Salt Lake County jail, Winder added.
Henderson's criminal history is limited to several misdemeanor drug-related offenses between 2009 and 2014, and a third-degree-felony weapons offense in 2014, according to a search of court records.
Reporters Matt Canham and Brennan Smith contributed to this story.