The Quakers' four-game winning streak is their longest since the end of the 2006-07 regular season.
Penn (9-8, 3-0), which remains tied for first in the Ivy League with Princeton, never trailed. The Quakers took a 38-24 halftime lead en route to their biggest league win since beating Columbia 87-55 on Jan. 14, 2006.
Eggleston scored 18 points and grabbed 10 rebounds for Penn, which shot 62.8 percent from the field (27 of 43), hit 10 of 16 3-pointers (62.5 percent) and outrebounded the Big Green 34-18.
Bernardini, who struggled all week with flu-like symptoms, scored 16 of his points in the first half, going 4 of 4 from 3-point range.
Kirk Crecco scored 13 for Dartmouth (5-14, 1-4).