Armstrong got the start in place of the injured Taylor Martinez, but playing time was evenly split between the redshirt freshman and Kellogg.
Nebraska (3-1), which finished with 645 yards, scored 38 straight points after falling behind 17-14 in the first quarter against the Championship Subdivision Jackrabbits (3-1).
Martinez didn't practice this week because of a turf-toe injury on his left foot and was in shorts and a T-shirt on the sideline. He had started 32 straight games since 2010.
The victory will do little to quell concern over Nebraska's defense. SDSU's Zach Zenner ran for 202 yards, tied for 10th-most by a Nebraska opponent, and the Jackrabbits finished with 465 total yards and 25 first downs.
Armstrong and Kellogg each played five series and threw one touchdown apiece. Armstrong led four touchdown drives and was 12 of 15 for 169 yards. Kellogg helped produce three TDs and a field goal and was 8 for 9 for 136 yards.
Abdullah, who scored the go-ahead touchdown on a 1-yard run early in the second quarter, didn't play after the Huskers' first series of the third quarter.
The victory ended a rough week for coach Bo Pelini and the Huskers. Nebraska blew an 18-point lead while losing 41-21 at home to UCLA last week, and Pelini got into a tiff with Husker quarterback great Tommie Frazier, who tweeted after the UCLA game that the defensive staff should be fired.
None of that compared to the firestorm created by the release of a 2-year-old audio of Pelini's profane rant against fans. The 2011 audio, leaked to the Deadspin.com sports website Monday, captured his off-air conversation with radio play-by-play announcer following a win over Ohio State.
Pelini apologized in a statement Monday and in a letter to fans posted on the school's athletic website Friday.
Fans greeted Pelini and the Huskers with their usual cheers and applause during the traditional Tunnel Walk, and he raised his right hand in acknowledgment as he walked off the field at the end of the game.
Those same fans were silenced in the first quarter as the Jackrabbits, ranked No. 6 in the FCS, moved up and down the field.
Zenner went untouched up the middle for a 40-yard touchdown and added a 3-yard TD before Justin Syrovatka kicked a 48-yard field goal to put SDSU up 17-14.
Zenner, a 2,000-yard rusher a year ago and a leading candidate for the Walter Payton Award as the top player in the FCS, went over 100 yards on just his seventh carry.
The Huskers created some cushion in the second quarter. Armstrong passed 16 yards to Sam Burtch for a 28-17 lead, and after Stanley Jean-Baptist's fourth interception in four games, Kellogg hit Tyler Wullenwaber to make it 35-17.
Mauro Bondi made it a 21-point game with a 22-yard field goal to end the half.
South Dakota State, which hosts two-time defending FCS champion North Dakota State at home next week, dropped to 0-6 against FBS opponents since elevating its program from Division II in 2004.