Moore completed his collegiate career by winning for the 50th time, the most victories for a quarterback in the history of major college football.
Boise State finished a 12-1 season and appeared in a bowl game for the 10th straight year.
Doug Martin opened and closed the Boise State attack, beginning the game by returning the kickoff 100 yards and producing Boise State's eighth touchdown on a 2-yard run with 2:24 to play.
Rashad Ross scored on a 98-yard kickoff return for Arizona State to start the second half, Jamar Taylor tallied for Boise State in the third period by bringing an interception back 100 yards and Travis Stanaway returned a fumble 26 yards for a Broncos touchdown in the fourth quarter.
Arizona State's George Bell finished the big-play game by catching a 30-yard pass from Brock Osweiler for a score with just 28 seconds remaining.
Martin's touchdown passes both came in the first half -- a 14-yarder to Tyler Shoemaker and a 2-yarder to Matt Miller.
Boise State scored the first 21 points and took a 28-3 halftime lead.
Arizona State won six of its first eight games this season, but closed the campaign with five straight losses and the school fired Coach Dennis Erickson.
Moore completed 26-of-34 passes for 289 yards and was intercepted twice.
Boise State, which lost only to TCU this season, was the fourth of seven Mountain West Conference teams to play in a bowl game this year. The conference has a 2-2 record this far in the bowl season.
Arizona State was the first team from the Pacific 12 to compete in a bowl, with five others to follow.