The Yellow Jackets scored touchdowns on three straight possessions of the third quarter, the last of which made it 35-7 on Allen's 5-yard run to the left side.
On the previous two drives, Allen ran down the right sideline for a 32-yard score, and Stephen Hill caught a 26-yard TD pass from Nesbitt.
Georgia Tech (5-2), which has a three-game winning streak, visits Atlantic Coast Conference rival Clemson next week.
Nesbitt needs 45 yards to break the ACC career record for yards rushing by a quarterback, a mark Clemson's Woodrow Dantzler set in 2001. His 1-yard run in the second quarter gave the Jackets a 14-0 lead and increased his ACC career lead to 33 for rushing touchdowns by a quarterback.
Middle Tennessee (2-4) took its second straight loss and third in the last four. The Blue Raiders host Louisiana-Monroe in Sun Belt Conference play next week.
Quarterback Dwight Dasher ran for two TDs, but Middle Tennessee's offense never regained its composure after he threw three interceptions -- two to Jackets safety Jerrard Tarrant -- on the Blue Raiders' first four possessions.
Georgia Tech made some mistakes, too.
In the first half, Scott Blair missed a 41-yard field-goal attempt, Allen lost a fumble and the passing attack suffered a few letdowns.
Nesbitt overthrew Hill on the Jackets' first offensive snap and missed a would-be touchdown. He missed another TD opportunity by overthrowing Embry Peeples on a deep route. Peeples dropped a TD pass, as did Correy Earls.
Allen, who finished with 75 yards on 13 carries, has scored five TDs in the last two games.
Orwin Smith put Georgia Tech ahead 7-0 with a 20-yard run.
Dasher's first TD, a 1-yard run, cut the lead to 14-7 late in the second quarter. His 6-yard run late in the fourth made it 35-14.
USC athletic director Pat Haden spoke Tuesday about the decision to fire head coach Steve Sarkisian, calling it "very difficult," while also defending his handling of the situation.
Steve Sarkisian has been plagued of late by apparent substance-related circumstances, including arriving to team facilities appearing intoxicated Sunday, according to sources.
Steve Spurrier, the visor-slinging, slick-talking "Head Ball Coach" who is the second-winningest coach in Southeastern Conference history, announced his resignation on Tuesday.
Florida Gators quarterback Will Grier has been suspended for the year after testing positive for a performance-enhancing substance found in a supplement, coach Jim McElwain announced Monday.
Tennessee defensive tackle Shy Tuttle will miss the rest of the season with a broken fibula and torn ankle ligament.
Former NFL safety Tyler Sash died from an accidental overdose after mixing two powerful pain medications, and a history of injuries was a contributing factor, a medical examiner said.