Reynolds set the record on a 4-yard run in the first quarter to give No. 20 Navy a 7-0 lead. He added three rushing scores later in the game to bring his season total to 17.
"When Coach called the play, I was like, 'I am not getting tackled,' " Reynolds said after the game. "I am scoring. I was tired of talking about it, when it's going to happen. I was ready to get it over with."
After the record-breaking score, Navy coach Ken Niumatalolo sent the team's equipment manager onto the field to secure the ball as a souvenir for Reynolds' keeping.
"I had a plan to make sure we did not lose that ball," Niumatalolo said. "I just wanted to make sure we got the ball to Keenan and his family. It was great for that to happen. I thought it was great he was able to present the ball to his dad. It's a pretty cool deal."
Reynolds finished the day with the four rushing touchdowns and 137 yards on 14 carries to go with 81 yards and a touchdown -- on a 79-yard pass -- on 2-for-9 passing.
The senior surpassed the mark set by former Wisconsin running back Montee Ball.
The NCAA record of 84 rushing scores belongs to Adrian Peterson, who played 42 games between 1998 and 2001 for Georgia Southern.
Information from The Associated Press was used in this report.