Willett shot a 67 to match the lowest round of the weekend, good enough for a 5-under 283 overall.
He was assured of his first major title when Spieth made bogey at the 17th hole.
Spieth had a five-shot lead at the turn but totally collapsed on the back side, dumping two balls in the water at No. 12 for a quadruple-bogey 7.
Information from The Associated Press was used in this report.