Eddie Lacy had 16 carries for 92 yards for Alabama and added 35 yards on four catches. Cooper had six catches for 136 yards a week after missing much the second half of the LSU game with an ankle injury, and failing to make a reception.
The Aggies had already lost to top-10 teams LSU and Florida by a combined eight points, proving they're already challengers in the powerhouse SEC.
"We played with a chip on our shoulder," receiver Ryan Swope said. "We were just trying to make a statement that we could compete with anybody in this league."
Manziel completed 24 of 31 passes with two touchdowns and ran 18 times, including four sacks.
He kept finding Swope. They hooked up 11 times for 111 yards and a 10-yard touchdown where Manziel bobbled the ball as defenders swarmed him, reversed field and spotted Swope alone in the back of the end zone.
The Aggies' first-quarter touchdown drives, giving them a 20-0 lead, stunned the Bryant-Denny Stadium crowd of 101,821. Christine Michael had a pair of 1-yard touchdown runs in the quarter, along with Swope's TD catch.
"We weren't stunned at all," Tide linebacker C.J. Mosley said. "As a defense, we knew they were going to make plays, that's what their offense depends on. They got a great quarterback, they got a great running back. We just have to settle down and play Alabama football."
The rest of the game was an emotional roller-coaster ride for fans who might have thought LSU was the toughest test on the road to another championship.
Then Alabama flexed its own muscle to counter Manziel's speed, sticking with power runs to set up 2-yard touchdown runs by T.J. Yeldon and Lacy.
McCarron also converted a fourth-and-4 with a pass to Lacy to set up the first score. Alabama marched back down the field after Mosley pushed Manziel out of bounds a yard shy on fourth-and-6.
The Tide ran out all but 19 seconds with a methodical drive right back into the game.
It was still the first time Alabama had trailed at halftime since a loss at South Carolina in 2010.
Jeremy Shelley kicked a 28-yard field goal with 4:49 in the third quarter to cut it to 20-17 and Taylor Bertolet answered with a short one to start the fourth.