Police have named him as Arcan Cetin, 20, of Oak Harbor, Washington. He was born in Turkey and is a legal US permanent resident, officials say.
The arrest took place in Oak Harbor, a town south-west of Burlington where the shooting happened.
Five people were killed inside Macy's department store at Cascade Mall.
The FBI said there was no evidence that the shooting was linked to terrorism.
At a news conference, police said Arcan Cetin had been identified as a suspect after a car connected to him was seen on security camera footage at the shopping centre.
Lt Mike Hawley, of Island County Sheriff's Department, said the car was later spotted in Oak Harbor. He said he had been on his way to the scene when he saw the suspect by the side of a road and arrested him.
Lt Hawley said the man "said nothing, he was zombie like" and was not carrying a weapon.
So far, police say they know of no motive for the shootings.
Police had launched a huge manhunt after the shootings, releasing a blurred photo of the suspect at the scene.
Three women and a girl were killed inside Macy's while a man later died in hospital.
The Seattle Times reported that the youngest victim was 16-year-old Sarai Lara, who had survived cancer as a young child.
Her mother described her as a happy, responsible, driven student.
Police say security cameras showed the suspect entering the shopping centre without a weapon. Ten minutes later the video shows the suspect entering Macy's with a rifle.
The gunman had fled the scene by the time police arrived.
A prayer service for the victims was held on Saturday evening at the Central United Methodist Church in Sedro-Woolley, Washington.
Five candles, one for each victim, were lit on a table in front of the church.
Burlington is about 65 miles (105km) north of Seattle.
A suspect sought over a fatal gun attack at a shopping centre in Washington state on Friday evening has been captured, police say.