The 37-year-old was two shots adrift of overnight leader Woods but kept his cool to card 70 to the American's 75 to win by three on the final green at the Hazeltine National Golf Club in Chaska, Minnesota.
Back came Wales and New Zealand finally lost their cool under pressure, replacement back-row Daniel Braid, on for the injured Kieran Read, earning a yellow card and presenting Jones with a regulation penalty to bring Wales to within a point.