The Tigers got the ball back with plenty of time, and Russell methodically led them down the field, milking the clock all the way.
On fourth-and-8, Russell got a first down with a completion to Doucet.
Russell appeared to score on a run from the Tennessee 7 when he flipped into the end zone, but officials ruled he was out of bounds at the 4. On the next play, a third down, Russell found Doucet in the end zone.
Davis finished 24-for-36 for 247 yards. He also ran for 71 yards.
LSU's sideline erupted and spilled onto the field. Order was restored for the kickoff, but Tennessee only had 5 seconds to score. Not enough.
The LSU players and coaches stayed afterward to celebrate with their fans. It was the Tigers' second win in Neyland Stadium, and they had to overcome four turnovers to earn it.
Last year at Tiger Stadium, the Vols overcame a 21-point deficit to win 30-27 in overtime.
LSU's top-rated defense made it tough for Crompton and the Vols' offense.
The Vols were held to 248 total yards and had the ball for less than 19 minutes. LSU rolled up 478 yards of offense.
Ainge started the game and played the first quarter but was limping and couldn't get much going. He sprained his right ankle in last week's game at South Carolina.
Crompton helped rally the Vols.
After LSU went ahead 21-17 at the end of the third quarter, the game turned wild with turnovers on three straight possessions in the fourth quarter.
Tennessee's James Wilhoit missed a field goal, and the Tigers got the ball back. But Russell had a pass intercepted, his third, by Antwan Stewart. Crompton had a mistake for the Vols three plays later, and LSU's Darry Beckwith got an interception.
The Tigers couldn't hold on. Dwayne Bowe caught a pass from Russell but was hit and fumbled. Tennessee's Ryan Karl recovered at the Vols 45.
On the next play, Crompton saw Meachem streaking for the end zone and threw it between two defenders. Meachem caught it and stretched his arm across the goal line.
Coming into the game Crompton had played in four games this season.
Keiland Williams' 7-yard touchdown run put LSU ahead 21-17 at the end of the third quarter.
The Tigers had cut the lead to 17-14 with 9:38 left in the third quarter on Bowe's 5-yard touchdown catch.
Bowe is now LSU's career leader in TD receptions with 22.
Tennessee went ahead 17-7 two plays into the second half when Russell overthrew a receiver, and Tennessee's Demetrice Morley caught it and ran 31 yards for the score.
Russell was shaken up on LSU's first possession when Tennessee's Xavier Mitchell grabbed his right leg and tried to pull him down. The play forced LSU to try a 52-yard field goal on fourth down, and Colt David's kick was short.
Backup Matt Flynn took over the Tigers' next series, but Russell returned for the next possession and led an 80-yard drive at the start of the second quarter. He completed a 23-yard touchdown pass to Buster Davis to go up 7-0.
Tennessee tied it with Crompton's 37-yard pass to Meachem, and Wilhoit added a 24-yard field goal to go up 10-7 at halftime.