Davies sank 10-of-11 foul shots and Tyler Haws struck for 24 points off the bench for the Cougars (23-11), who bested Washington in the first round.
Matt Carlino went 5-of-10 from 3-point range en route to 18 points and six rebounds while Brock Zylstra added 14 and six as BYU advanced to take on top-seeded Southern Mississippi in the next round.
Mercer (24-12) was paced by Travis Smith with 19 points.
Langston Hall added 15 for the Bears.
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