Moe turned a short sideline pass into a touchdown with a stutter step that caused two Aztecs defenders to collide near midfield. Missouri (3-0) had fallen behind by four points after a 32-yard field goal with just over two minutes remaining for San Diego State (2-1).
The dramatic win offset a career day by San Diego State freshman Ronnie Hillman, who ran for 235 yards and touchdowns covering 75 and 93 yards. He overshadowed his first-year counterpart Henry Josey, who had 44 yards on 10 carries for a Missouri team that totaled just 89 yards rushing.
Gabbert ran for a first-half touchdown and was 28 of 51 for 351 yards with two interceptions, both in the final five minutes. Tight end Michael Egnew caught 13 passes for 145 yards, with Moe collecting seven for 119 yards.
A 15-point favorite entering the game, Missouri appeared to be in control after Kevin Rutland's goal-line interception following a Tiger fumble preserved a 20-14 lead early in the fourth quarter.
But Memorial Stadium grew quiet after Hillman's full-field burst gave the Aztecs a one-point lead just two minutes later. Hillman's first long TD run with 35 seconds remaining in the first half similarly silenced the home crowd, trimming what had been a 10-point Missouri lead to 17-14.
After the Aztec running back's second long score, a smattering of boos could be heard when Gabbert was stopped on a first-down quarterback keeper for no gain with 8:25 left.
That changed after an electric score by Moe, who entered the game tied for the most catches in the country but was overshadowed for most of the night by his teammate Egnew.
"He knows he's good now," a relieved Missouri coach Gary Pinkel said of Moe.
San Diego State, off to its best start since 1994, had hoped to snap a 22-game losing streak against teams from a Bowl Championship Series conference.
The Aztecs entered the game ranked sixth in the country in total offense and collected 440 yards, the same total as Missouri. But they also were flagged for 10 penalties -- compared to just two for Missouri -- including three delays of game.
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The silence from LSU's administration this week about Miles' job status spoke volumes, but not as loudly as the support Tigers fans showed on Saturday. It's been a strange week in Baton Rouge.
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Donnel Pumphrey and Chase Price surpassed 100 yards rushing on Saturday night, and San Diego State completed an undefeated conference season with a 31-14 win over Nevada.