The Tigers have won four straight games, averaging more than 53 points per contest during the streak.
Witter and Rountree both scored, as Missouri amassed 433 yards on the ground. Rountree had a 64-yard run to the Tennessee 8 to set up his own touchdown, which put the Tigers (5-5, 2-4 SEC) ahead 24-17 with 18 seconds left in the second quarter.
In the second half, Missouri quarterback Drew Lock threw touchdown passes to Emanuel Hall, J'Mon Moore and Johnathon Johnson, and the Tigers defense shut out Tennessee.
Much of the Tigers' resurgence has been due to the play of Lock.
He leads the nation with 35 touchdown passes, including 18 over the past four games.
On Saturday, Lock was 13-for-28 for 217 yards and four touchdowns. Lock also threw an interception that was returned for a touchdown.
Freshman quarterback Will McBride, making his first start for the Volunteers (4-6, 0-6), completed 16 of 32 for 139 yards and one touchdown. He was picked off twice.
McBride, who started the season as Tennessee's third-string quarterback, made his debut a week ago when Jarrett Guarantano was injured in the second quarter of the Volunteers' 24-10 win over Southern Mississippi.
Tennessee converted a fourth-and-2 from the Missouri 26-yard line to set up a 19-yard touchdown pass from McBride to Ethan Wolf that tied it late in the first half.
But the Tigers needed just 57 seconds to set up Rountree's score.
Rountree and Witter combined for 219 yards rushing for the Tigers in the first half.
Nigel Warrior intercepted a Lock pass and returned it 70 yards for a touchdown to tie the game at 7 midway through the first quarter.
Missouri has road games versus Vanderbilt and Arkansas remaining as the Tigers try to become bowl-eligible for the first time since going 11-3 in 2014.
The Volunteers need to win their last two, at home games against LSU and Vanderbilt to have a shot at a bowl game.