South Alabama (2-2) had first-and-goal at the 7 in the closing minutes but was unable to get the tying touchdown. Facing fourth-and-goal from the 8, South Alabama quarterback Ross Metheny was hit by Corey Vereen while releasing the pass that Randolph intercepted.
Tennessee's Rajion Neal rushed for a career-high 169 yards on 25 carries. Neal's 11-yard touchdown run gave Tennessee (3-2) a 31-7 lead early in the third quarter.
Metheny was 21 of 42 for 234 yards with two interceptions. He also rushed for 67 yards and two touchdowns.
Tennessee managed to avoid a third straight loss as it heads into a brutal three-game stretch. The Vols host No. 9 Georgia and No. 12 South Carolina and visit No. 1 Alabama in their next three games.
The Vols overcame a three-interception performance from Justin Worley, who was 20 of 36 for 234 yards. Worley threw touchdown passes to freshmen A.J. Branisel and Josh Smith, but he also threw two interceptions into the end zone late in the first half to prevent Tennessee from extending its lead.
Tennessee still led 31-7 early in the third quarter, but the Vols struggled to move the ball the rest of the game while South Alabama gradually chipped away at the lead
South Alabama also capitalized on Tennessee's mistakes.
After a false start penalty on Tennessee's Alex Bullard nullified a successful fake field-goal attempt with the Vols leading 31-17, Michael Palardy's ensuing 52-yard field goal try was about 15-20 yards short. South Alabama's Roman Buchanan caught the errant kick and returned it to the Tennessee 29.
Cris Dinham capped South Alabama's ensuing possession with a 3-yard touchdown run that cut Tennessee's lead to 31-24 with 8:55 remaining.
South Alabama had shown it wouldn't be intimidated by the Neyland Stadium atmosphere by scoring a touchdown on the game's opening possession. Metheny went 4 of 4 for 55 yards on that opening series and capped the drive with a 32-yard touchdown run.
Tennessee regained control later in the first half.
Neal had a 53-yard run and Marlin Lane added a 54-yard burst to set up touchdowns as Tennessee reached the end zone on three straight first-half possessions. Lane's run was the Vols' longest play from scrimmage this season.