Miami (11-4, 1-2) had its 21-game home winning streak snapped.
The Fighting Irish were down 61-57 with 2:30 left, but Beachem's reverse layup put them ahead 63-61 with 1:04 to go. After Ja'Quan Newton made one of two free throws for Miami, Steve Vasturia drove for a layup with 8 seconds remaining.
Following a timeout, Beachem made a steal on the inbounds play, and Matt Farrell's two free throws iced the victory.
Notre Dame won despite shooting a season-low 37 percent, with Miami blocking a season-high 12 shots.
Farrell led the Irish with 15 points and six assists, and Beachem had 13 points.
Miami senior Davon Reed tied a career high with 21 points but made a critical turnover down the stretch. The Hurricanes had 13 turnovers to only six for Notre Dame.
The Fighting Irish, who outscored opponents 25-6 in the final 6:42 of their first three league games, again came through down the stretch. Their other conference wins were by one, seven and five points.
The Irish led only briefly in the first half but outscored Miami 10-0 to begin the second half and take a 38-28 lead. Bonzie Colson started the run with a 3-pointer, and Martinas Geben finished it with a dunk.
Notre Dame had its biggest lead at 50-39 before the Hurricanes went on a 13-2 run to forge a tie and set the stage for the frantic finish.
Notre Dame will remain on the road to play at Virginia Tech on Saturday. That's the second of three consecutive away games for the Irish.
Miami plays at Pittsburgh on Saturday, and its next three games are on the road.