As the (self-appointed unofficial) Bama historian here, a couple of things about this.
1) Dickey Moegele (it was pronounced "May-gull," quickly) was a running back for Rice University. (He later changed the spelling because it kept getting mis-pronounced, so technically both of you are correct). (Keep in mind most players were two-way players back in that day and freshmen weren't allowed to play). Tommy Lewis was a Bama fullback who was on the sidelines.
2) Moegele set a Cotton Bowl record for rushing that day that stood until it was broken by Missouri RB Tony Temple in 2008.
3) Rice University was a powerhouse in the 1950s (sorta like TCU and some others you just don't think of). In 1953, they won the Southwest Athletic Conference title with a 9-2 record (they lost to 5-5 SMU and a 7-2-1 Kentucky squad coached by YOU KNOW WHO in his last game before departing to aTm). So Rice drew the Cotton Bowl. (It was also the last bowl game for Red Drew, who would retire after the 1954 season). Also of note, Bart Starr was the Bama QB.
4) Moegele had one of those "put the team on my shoulders and carry them" days, dashing for 265 yards and three TDs: 79, 95, and 34 yards. It was the 95-yard dash that he never completed. Lewis jumped off and tackled Moegele. I have the HBO "Not So Great Moments In Sports" where Moegele said that he saw Lewis at the last second and tried to avoid him (you can see him swerve slightly on the video), but Lewis took him down. Lewis apologized profusely after the game, and Moegele told him to forget about it, it was no big deal. They wound up on "The Ed Sullivan Show" together, and Moegele even admitted to being a little stunned by the fact that somehow Lewis came out of the thing looking like a hero when he should have been the goat.
And before you ask, I know it has happened at least twice more since then. In 1984 it happened in one of those small games (it involved either UConn or Nevada IIRC), and I think it was a coach. It also happened sometime late in the 1990s. Ironically, nobody even remembers those for the most part.