The Scarlet Knights' second shutout of the season made them bowl eligible on the earliest date in school history.
The last Rutgers team to start 6-0 was the 1976 squad, which finished 11-0.
With Hampton out, the Scarlet Knights limited the nation's top-ranked rushing offense to 113 yards on 50 carries. Navy (5-2) had been averaging 350.5 yards on the ground and had not been shut out since a 38-0 defeat against Connecticut in 2002.
"They whipped our tail. We couldn't run or throw the ball," Navy coach Paul Johnson said.
Hampton, the key to the triple-option offense, was hurt on a tackle by Ramel Meekins at the end of a 5-yard run.
"I feel bad for Brian. He was really starting to play well and grow into the position," Johnson said. "A lot of guys had tears in their eyes. It really affected some of them."
As Hampton was lifted to the stretcher, many of the 36,918 fans chanted his name. Hampton responded to the largest crowd in the history of Navy-Marine Corps Stadium by raising his arm with his index finger extended.
"I was on the field trying to calm him down, and all he did was keep apologizing to me," Johnson said. "There wasn't anything he could have done."
Hampton was replaced by sophomore Kaipo-Noa Kaheaku-Enhada, who had thrown only five passes in Navy's first six games.
Kaheaku-Enhada rushed for 11 yards on 22 carries, numbers that included several sacks. He completed two of six passes for 28 yards before being replaced by Jarod Bryant with 13 minutes left.
"You can't lay all the blame on the quarterback," said Navy fullback Adam Ballard, who finished with 23 yards on eight carries. "We didn't do anything right. It was a complete meltdown."
"If one breakdown occurs, they can have a big play. All of a sudden the fullback comes jetting out of there for 60 yards," Schiano said. "So you never feel comfortable."
Teel, a sophomore, went 15-for-26 for 215 yards. Tiquan Underwood caught two touchdown passes and Ray Rice ran for 93 yards on 21 carries, the first time this season he was held under 100 yards.
The Scarlet Knights made it 10-0 late in the half with a six-play drive that covered 59 yards in 38 seconds. Three plays after a Teel completed a 29-yard pass to James Townsend on a third-and-10, Underwood caught a short pass over the middle and took it in for a 25-yard score.
A blocked punt in the third quarter set up a 9-yard touchdown pass to Underwood for a 17-0 lead. The Midshipmen were then forced to punt, and Glen Lee blocked Greg Veteto's kick and recovered it to the Navy 9.
That led to a field goal, and Rutgers went up 27-0 early in the fourth quarter on a 30-yard touchdown pass from Teel to Townsend.
Navy's first possession nearly ended in disaster. Hampton fumbled upon being sacked, and Rutgers' William Beckford took the ball 6 yards to the Navy 1 before he fumbled into the end zone. The Midshipmen recovered for a touchback.