Both players finished 72 holes at 6-under 274. Roberts ballooned to 5-over 75 in the final round while Romero shot 1-over 71.
Roberts parred the first playoff hole to earn his fifth Champions Tour win.
"I really feel relieved, I've always had a lot of faith in my putter," Roberts said. "I hated it to end that way for Eduardo, but it feels wonderful to be champion. I've worked a lot of years to become a major champion."
Dick Mast, who shot the low round of the day with a 3-under 67, was one of five players to finish in red figures. He was third at 5-under 275.
Roberts won $293,981 for the victory.