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Grigsby raced 94 yards to the Lumberjacks' 1-yard line early in the third quarter. That was two yards shy of a school-record 96-yard run by Trung Canidate in 1997.
Grigsby opened the scoring with a 25-yard touchdown run five minutes into the game and went 30 yards for another score six minutes later.
The Wildcats (2-0) have won four straight games for the first time since 2000.
Arizona's first two wins this year have come against Central Michigan and NAU (0-1). Next week, the Wildcats invade Big Ten country for a game against Iowa, coach Mike Stoops' alma mater.
Northern Arizona made the Wildcats work.
Down 14-3 early in the second quarter, the Lumberjacks marched 79 yards in 14 plays, scoring on a 4-yard pass from Michael Herrick to Ed Berry to cut the deficit to 14-10.
That's how it stood until Arizona's Matt Scott hit Terrell Turner for a 2-yard score with 30 seconds to go in the half. Making his second college start, Scott completed 14 of 20 passes for 150 yards and a touchdown, and he threw his second interception in as many games.
Cornerback Trevin Wade had two interceptions for the Wildcats, who are 10-1 against Northern Arizona.
The first came near Arizona's goal line with the Wildcats leading 21-10 early in the third quarter. On the next play, Grigsby went 94 yards to the NAU 1-yard line, setting up a touchdown plunge by Keola Antolin.
Wade picked off another pass later in the third quarter.
Backup quarterback Nick Foles, a sophomore transfer from Michigan State, made his first appearance with Arizona leading 27-10 midway through the third quarter.
Foles completed 6 of 8 passes for 44 yards and a touchdown. He also lost a fumble.
Scott returned in the fourth quarter and threw an interception to Matt Steinbach, setting up a 19-yard touchdown pass from Herrick to Berry.
Herrick completed 20 of 30 passes for 159 yards and two touchdowns. He was intercepted twice.
