Snell rushed for 122 yards and pair of touchdowns as Kentucky survived a game that went down to the game's final play.
Missouri looked better than it has this season. The Tigers tied the game at 34 with 13:46 left in the fourth quarter after a 75-yard touchdown pass to Johnathan Johnson. Quarterback Drew Lock had previously connected for touchdowns of 58 yards to Emanuel Hall and 50 yards to J'Mon Moore.
But his final attempt from 27 yards on the game's final play fell short. Lock finished with 355 yards and three touchdowns.
His Kentucky counterpart Stephen Johnson was equally impressive. The junior threw for 298 yards and two touchdowns. Of those yards, 111 of them went to Garrett Johnson, who added a touchdown, as did teammate Blake Bone.
The game ultimately came down to Wildcats kicker Austin MacGinnis, who made two field goals (53 and 20 yards) in the fourth quarter.
Kentucky moves to 5-1 overall and 2-1 in the SEC, while Missouri is 1-4 and 0-3.
Mizzou claimed 568 total yards to Kentucky's 486 yards.
Lock scored Missouri's first rushing touchdown against an FBS opponent this year.
Kentucky's Luke Wright was taken to a local hospital after collapsing on the sideline.