Nine people were killed and two injured in the shootings, which began in west Phoenix's Maryvale neighborhood.
"They had a clear mission: bring this killer to justice and get it right".
Gisela Castro, the mother of shooting victim Manuel Castro-Garcia, said news of the arrest felt like she was reliving the day she was told her son had been killed.
"We hope that our community will rest a little easier and that our officers will get a little more sleep knowing that the wheels of justice are finally in motion", Police Chief Jeri Williams said.
Saucedo was taken into custody last month and charged with the August 2015 murder of Raul Romero, 61 - a homicide that hadn't yet been linked to the pattern.
Investigators initially believed a serial shooter was responsible for seven deaths a year ago, but in recent weeks they determined two other victims were also shot by the same gunman, who they believe to be Saucedo, including the victim killed in August 2015.
"I'm really just shocked", she said.
An attorney for Saucedo did not return a call seeking comment.
Police arrested Saucedo last in connection with a separate 2015 killing.
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The relatives of two victims in a serial shooting case that Phoenix police say they've solved say they're relieved but hurting emotionally.
Police said the killings were unrelated to a series of 2015 freeway shootings in Phoenix that terrified drivers.
The killer was dubbed the Serial Street Shooter.
"The fact of them finding somebody, at least it was some type of news", Smith said.
The Maricopa County prosecutor said his office hadn't yet decided which charges they would bring to trial. Last week, Saucedo pleaded not guilty to that shooting. But they said in a statement the evidence includes surveillance video, witness statements and ballistics material.
Jesse Olivas, 22, was shot dead on the street on January 1, 2016.
Police also have linked Saucedo to the April 19, 2016 death of 60-year-old Krystal White; the June 3, 2016, death of Horacio de Jesus Pena; and to the killings of two women, Angela Linner and Stefanie Ellis, and 12-year-old girl Maleah Ellis on June 12, 2016.
Nobody was killed in those shootings, in which about a dozen vehicles on the interstate outside the city were shot between August 29 and September 10.
Shots were fired into a vehicle on July 11, 2016; a man and a 4-year-old boy inside escaped unhurt.