Both players ended regulation with scores of 15-under-par 201.
In the final round, Bean shot 4-under 68, while Eaks shot 7-under 65 to force the playoff.
Bean had to birdie the last hole. It was his first Champions Tour win.
"It's been a long time," said Bean, who won $240,000. "I've come close and not done it, but it feels good. It's a relief. The money is nice, but the trophy is what I wanted. Your name stays on the trophy forever."
Chip Beck (67) finished alone in third place at 13-under 203.
Dana Quigley (66) finished in fourth place at 12-under 204.