Tre Mason's 5-yard score with less than 2 minutes to play was first placed at the 1, but it was reviewed and ruled a touchdown.
A&M (5-2, 2-2 SEC) had a last chance, but Manziel, the Heisman Trophy winner, was sacked by Dee Ford on fourth down to secure the win.
Manziel threw for 454 yards and four touchdowns and ran for a fifth score, but was intercepted twice. He missed a series in the fourth quarter with an apparent injury to his right arm or shoulder. He returned but was constantly moving his right arm.
Marshall threw for 236 yards and two touchdowns and ran for 100 yards and two more scores for Auburn (6-1, 3-1), which lost 63-21 to the Aggies last season.
Texas A&M's Mike Evans had a school-record 287 yards receiving with four touchdowns to become the first player in school history to have two, 200-yard receiving games in a career. He scored on receptions of 26, 64, 42 and 33 yards.
Auburn entered the game leading the SEC with 287 yards rushing a game and piled up 379 yards rushing against the Aggies. Mason led the way with 178 yards and Corey Grant added 45.
Manziel connected with Evans for his fourth TD on a 33-yard pass - that Auburn was called for pass interference on - giving A&M a 31-24 lead late in the third quarter.
Manziel was injured when he was tackled on an 8-yard run early in the fourth quarter.
He got up after the tackle and then went to the ground before he reached the A&M sideline. A&M officials looked at him for a couple of minutes on the turf before he got up and walked off on his own power.
He was replaced by Matt Joeckel, who threw an incomplete pass on third down and A&M settled for a 20-yard field goal to push the lead to 34-24.
Marshall cut A&M's lead to 3 when he ran 13 yards for a touchdown with about 12 minutes left.
Joeckel remained in for A&M's next possession and the Aggies had to punt.
Cameron Artis-Payne put Auburn ahead 38-34 with a 2-yard run with about 9 minutes remaining. Tre Mason slashed through the line and ran 53 yards to set up the score.
Manziel returned and orchestrated a 75-yard drive capped by his 1-yard touchdown run to give A&M a 41-38 lead with about 5 minutes left.
A&M had forced Auburn to punt on four straight possessions before Marshall found Sammie Coates on a screen pass and he coasted into the end zone untouched for a 43-yard touchdown to tie it at 24-24.
Evans took a short pass from Manziel and darted through traffic for a 26-yard touchdown that gave A&M a 7-0 lead in the first quarter. It was the sixth straight game the Aggies scored a touchdown on their opening drive.
The Tigers answered when Marshall ran 14 yards for a touchdown on Auburn's first drive. Grant had a 32-yard run two plays before the touchdown.
Manziel's first interception came when Ryan Smith grabbed a pass that was deflected by Nehemiah Hicks just before it hit the ground in the first quarter. Auburn took a 10-7 lead on a 27-yard field goal by Cody Parkey after the interception.
Texas A&M's next drive was a quick one. Evans grabbed an 11-yard pass before taking a short throw, getting free on a block by Malcome Kennedy and dashing 64 yards down the sideline for a touchdown that put A&M back on top 14-10.
The Tigers coughed the ball up on their next possession when Howard Matthews forced a fumble by Marshall which was recovered by Steven Jenkins.
But the Aggies weren't able to take advantage of that miscue, with Manziel throwing another interception six plays later. He evaded three defenders before launching the pass into double coverage which was intercepted just in front of the goal line by Ryan White.
Auburn had a first-and-goal at the 1 after converting a fourth down after the interception by Manziel. But the Tigers were flagged for a false start before Marshall was sacked for a 7-yard loss to bring up third down.
Marshall escaped the rush and found Quan Bray in the end zone for the 13-yard touchdown that gave Auburn a 17-14 lead early in the second quarter.
A&M tied it with a 37-yard field goal by Josh Lambo.
The Aggies regained the lead just before halftime when Evans scored again. This time he pulled in a short pass, juked a defender and then stiff-armed another before diving into the pylon for a 42-yard touchdown to make it 24-17.