The Tigers (9-2, 6-1 Southeastern Conference) set a school record with their 10th straight road win.
Missouri can clinch the East Division and earn a second straight trip to the SEC championship game by winning at home Friday over Arkansas. A loss would send No. 9 Georgia (9-2, 6-2) to Atlanta instead. Georgia is 10th and Missouri 20th in the College Football Playoff standings.
Andrew Baggett's 43-yard field goal broke a 13-all tie midway through the third quarter. Mauk threw a 73-yard touchdown pass to Jimmie Hunt and a 13-yard scoring strike to Bud Sasser in the fourth quarter.
Tennessee (5-6, 2-5) must win next week at Vanderbilt to become bowl-eligible for the first time since 2010.
Joshua Dobbs threw a 4-yard touchdown pass to Jason Croom with 1:52 remaining and delivered a two-point conversion on a quarterback keeper. Tennessee then rested its hopes on an onside kick.
Tennessee recovered its first attempt, but an offside penalty forced the Vols to kick again. Although Tennessee's Justin Coleman recovered the next kick, officials ruled the Vols had touched the ball before it went the required 10 yards.
Missouri won by containing a Tennessee offense that had averaged 47.5 points per game in Dobbs' two previous starts. Missouri had six sacks and kept Tennessee's offense out of the end zone until the final 2 minutes. The Volunteers' only touchdown before that came when holder Patrick Ashford threw a 31-yard pass to Alex Ellis on a fake field goal.
Tennessee was playing without leading tackler A.J. Johnson and cornerback Michael Williams, who were suspended from all team-related activities Monday, one day before both were named as subjects of an ongoing rape investigation in which no charges have been filed.
The Vols also didn't have injured center Mack Crowder and receiver Marquez North and played the first half without safety Brian Randolph, who had committed a targeting penalty in a 50-16 victory over Kentucky last week.
All those absences tested Tennessee's depth to the point that the Vols started seven true freshmen. Never before had Tennessee started that many true freshmen in one game.
Tennessee still seemed on the verge of tying the game or taking the lead early in the fourth quarter.
The Tigers were clinging to a 16-13 advantage when a pass-interference penalty gave Tennessee a first down at the Missouri 29. But on the next play, Dobbs threw a pass that bounced off the hands of wide receiver Josh Malone and was picked off by Kenya Dennis at the Missouri 20.
Three plays later, Mauk threw a deep pass to Hunt, who beat Coleman to catch the ball inside the Tennessee 40, then stiff-armed cornerback Cam Sutton inside the 25 before heading into the end zone. The touchdown extended Missouri's lead to 23-13 with 10:42 remaining.
Mauk capped Missouri's next drive by finding Sasser in the left corner of the end zone with 6:22 left.
Butch Jones post-game