Freshman quarterback Justin Fields scored on runs of 12 and 15 yards, and Isaac Nauta had a 31-yard touchdown on a fumble return. The Bulldogs led by only 12 points early in the final quarter before pulling away.
Georgia (5-0, 3-0 Southeastern Conference) scored touchdowns on its last drive of the first half and opening possession after halftime but otherwise couldn't maintain momentum.
Jake Fromm completed 16 of 22 passes for 185 yards while sharing time with Fields.
Tennessee (2-3, 0-2) was held to 209 yards. The Vols finally scored on Jarrett Guarantano's 37-yard pass to Josh Palmer late in the third quarter. Guarantano's 35-yard scoring pass to Ty Chandler about four minutes into the final quarter cut Georgia's lead to 24-12.
Georgia answered with a 75-yard drive capped by Swift's strong 14-yard scoring run. The Bulldogs took advantage of Tennessee's first turnover, a fumble by Jeremy Banks, to stretch the lead on Fields' 15-yard scoring run.
"When we had to have a drive in the second half, I thought we came up with a drive," said Georgia coach Kirby Smart, who said running back Brian Herrien "had some juice" to spark the offense.
Coach Jeremy Pruitt was encouraged even though the loss extended Tennessee's school-record losing streak in conference games to 11 since a 63-37 win over Missouri on Nov. 19, 2016.
"I learned we've got fight in us, that we're not going to quit," Pruitt said. "It kind of gets me excited. I'm proud of them and the way they fought. Our guys kept fighting, which means we're headed in the right direction and that excites me. That's why I'm getting a little bit emotional."
Guarantano, who left last week's loss to Florida after taking a hit to the left knee, wore a brace on the knee while completing 13 of 21 passes for 143 yards and two touchdowns.
The Vols failed on 2-point conversions after each touchdown.
The bad news for Georgia's misfiring offense was two first-half fumbles by Fromm. The good news is the first fumble, forced by a hit from linebacker Darrell Taylor, was returned 31 yards by Nauta for a touchdown to cap the Bulldogs' opening drive.
Fromm said he had "no idea" what happened after he lost the ball.
"I stood up and heard the crowd cheer and was like, 'What the heck is going on?’ ” Fromm said. "... I had to look up at the board and see what the heck happened."
Fromm fumbled again in the second quarter, and center Lamont Gaillard's recovery led to a 43-yard field goal by Ricardo Blankenship.
Swift had 50 yards rushing in Georgia's balanced running attack led by Elijah Holyfield's 16 carries for 78 yards.
Tennessee: In a sign of improvement, after six turnovers last week against Florida and a combined eight the last two weeks, the Vols had no turnovers before Banks' fumble was recovered by Georgia's Juwan Taylor with 3:24 remaining. Another reason for encouragement was the comeback attempt that cut the deficit to 12 points early in the final quarter before the Vols faded.
Georgia: The Bulldogs showed improvement in their run defense. After giving up 172 yards on the ground in last week's 43-29 win at Missouri, Georgia held Tennessee to 66 yards rushing, including 30 yards on 12 carries in the first half.
"We knew we couldn't let it happen again," said linebacker D'Andre Walker, referring to the Missouri game. "That wasn't our brand of football."
NO QB PLAN
Smart fielded more than one question about his plan for his quarterbacks after Fields' bigger role. The coach repeated the same response three times: "The plan is there is no plan."
Tennessee receiver Marquez Callaway did not return after a hard hit on his 13-yard catch early in the second half. ... An official, field judge Sean Petty, was carted off the field after he suffered a left leg injury on Swift's scoring run late in the game.
LEGEND VS LEGEND
In a matchup of team greats, Georgia's Herschel Walker accompanied the team on its pregame Dawg Walk into the stadium and led a pregame cheer on the field. Meanwhile, former Tennessee quarterback Peyton Manning was on the Vols’ sideline.
Tennessee: Following a week off, the Vols face another ranked SEC team when they visit No. 10 Auburn on Oct. 13. Tennessee plays No. 1 Alabama on Oct. 20.
Georgia: The Bulldogs play their fourth game against a team from Tennessee when Vanderbilt visits Sanford Stadium next Saturday. Georgia already has home wins over Austin Peay and Middle Tennessee State.