Tyler Haws added 24 points, and Matt Carlino had 18 points and nine assists for the Cougars (23-11).
Travis Smith scored 19 points for Mercer (24-12) and Langston Hall added 15. Smith and Hall combined to hit 8 of 13 3-point attempts.
After Smith's 3-pointer to open the scoring gave the Bears their only lead of the game, the Cougars scored the next 11 points behind two 3s by Carlino. BYU went up 28-11 when Carlino hit the third of his five 3s.
"When you start a game like that it allows you to get a little more aggressive defensively," BYU coach Dave Rose said.
Mercer hit only 3 of 13 shots to open the game but then made 11 of 15. The Bears scored 10 straight to cut the margin to four but trailed 44-35 at the break when Carlino hit another 3-pointer.
"I thought he was really special," Mercer coach Bob Hoffman said of BYU's sophomore point guard. "He took the challenge and was really good. ... I thought he was as good as we've seen anywhere, any time at any place."
A 3-pointer by Hall cut BYU's lead back to four early in the second half, but the Cougars put together a 15-3 spurt to go up 80-63. Combined, Carlino and Brock Zylstra were 9 of 18 on 3s for BYU. Hoffman said BYU's long-range shooting stretched the Bears' defense.
"Our bigs were trying hard, but [Davies] is a talented guy who knows how to get where he wants to get. It's hard to get down in there to guard him when those guys are hitting 35-footers," Hoffman said.
BYU hit 22 of 29 free throws to Mercer's 8 of 14 from the line. The Cougars outrebounded the Bears 43-31.
"We gave them a couple of runs and [the Bears] came right back at us again," Rose said. "That's a sign of a championship team. They don't give up and they don't back down."
The Cougars advance to the quarterfinals to play Southern Mississippi, winners over Louisiana Tech on Monday. Davies, a senior, played his final game at home.
"We're rolling and it's a lot of fun. It's like everyone is dialed in," he said.
Mercer entered the game giving up only 58 points per game. The 90 points BYU scored are the most the Bears have allowed this season.