Brock Zylstra led the Cougars with 23 points. Davies finished with 20 and Rogers ended up with 19.
Griffon Jones scored a game-high 26 for Dixie State, which owned a 78-77 lead before Davies and Rogers went to work.
"Hats off to them," BYU's Noah Hartsock said. "... They gave us a great challenge."
Davies' thunderous rebound dunk gave the Cougars an 81-78 lead with 3:32 remaining.
With 1:04 remaining, Davies was intentionally fouled after another offensive rebound.
His made two free throws to make it 86-81 and, with BYU still in possession, Rogers launched a quick three-pointer that probably wasn't the best shot except he made.
After Dixie's Dalton Groskreutz quickly scored, Rogers struck again. His three-pointer gave the Cougars a 92-83 lead with 51 seconds remaining.
For those keeping track, Rogers scored six points in nine seconds.
Hello, Jimmer Fredette.
"Ever since I got here," Rogers shrugged, "the coaches have told me to keep shooting."
Said Rose: "What concerns me is we won the game offensively. ... We've been better defensively than we are right now."
Exhibit A: Dixie State shot 51.6 percent against the Cougars, including 58 percent in the second half.
The Red Storm also went 8-for-15 from the three-point line. Solomon Jensen was 3-for-6 and finished with 14 points. He was one of four Dixie players to reach double figures.
"We lacked attention to detail," Hartsock said. "... We were just out of sync defensively. [But] we were able to make winning plays at the end. That's something good we can take from this game."
R Brock Zylstra scores 23 points and BYU beats Dixie State, 96-85, in an exhibition game Friday night.
• The Cougars outscore the pesky Red Storm, 19-7, in the final four minutes of the exhibition game.
• Griffon Jones scores 26 points for Dixie, which jumped to a 12-2 lead in the first three minutes.