Clint Mann chipped in with 12 points and six rebounds as Davidson grabbed 20 offensive boards in the win.
Davidson (21-7, 15-2 Southern Conference) led by just three, 31-28, at halftime, but opened the second half on a 22-9 run. Brooks capped the run with two free throws as the Wildcats took a 53-37 lead with 11:33 left to play.
Brooks scored 15 points during the run and Mann added six.
Davidson outrebounded Elon 50-35 and forced the Phoenix to turn the ball over 15 times.
Lucas Troutman paced Elon (13-15, 8-9) with 15 points but no other Phoenix scored more than six points.
Davidson leads the all-time series 34-13 and have won the last two matchups against the Phoenix.