Doug Ducey embraced McCain's wife, Cindy, when the motorcade arrived.
Senator Jeff Flake offered the benediction at the service.
Wednesday's ceremony inside the Arizona State Capitol marked the beginning of McCain's week-long journey to his final resting place in Annapolis.
McCain died on Saturday at the age of 81 and Donald Trump offered no honest condolences on the day of McCain's death.
McCain's remains were driven to the state capitol in a black hearse escorted by four policemen on motorcycles.
U.S. Sen. Jerry Moran, R-Kan., delivered a speech this week on the Senate floor in which he recalled meeting McCain through Dole in 1996.
During the CBS interview, Graham criticized Trump for belatedly issuing a proclamation for USA flags to fly at half-staff until McCain's burial this weekend.
McCain's body arrived in a black vehicle and was greeted by uniformed personnel lining the street, leading up to the Capitol.
Veteran Judith Hatch handed out flags to those assembled, saying Arizona lost a champion for the military.
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"We were the ones who were privileged", said Ducey.
Kyl said many people can disagree with McCain's political views, but they should not diminish their gratitude for his service.
Former Vice President Joe Biden is among speakers at the church service, which will be attended by dozens of senators, other politicians and some professional athletes. Jon Kyl who both gave heartfelt remarks.
McCain and Trump also had a deeply strained relationship, so much so that the president at first refused to keep federal flags lowered to half-staff until McCain's burial on Sunday. Along with invited family and friends, around 1,000 seats were being made available to members of the public who signed up.
Political giants, local athletes, and the people who elected him to serve Arizona for over three decades in Congress will again honor the senator.
Brewer described McCain's wife, Cindy, as "very composed" as she approached the casket and put her cheek against the flag. The two others were state Senator Marilyn Jarrett in 2006 and Olympic gold medalist Jesse Owens, a Tucson resident, in 1980.
Arizona Cardinals' receiver Larry Fitzgerald is scheduled to will speak during the service in Phoenix, preceding former USA vice-president Joe Biden. Musical choices include a performance of "Amazing Grace" by the Brophy Student Ensemble and a recessional to "My Way" by Frank Sinatra.
Following a service at North Phoenix Baptist Church, McCain's casket will go to Phoenix Sky Harbor International Airport for a salute from the Arizona Air and Army National Guard.