Williams’ shot from the edge of the area, after a clearance from a corner, took a wicked deflection for an 88th-minute equaliser before Wood tapped home 60 seconds later after a swift counter-attack set up by inspirational captain Michael Bradley.
It was the first time the States had beaten the Dutch, whose defeat continued a run of mediocre results since they came third at last year’s World Cup. It will also heap the pressure on them ahead of next week’s vital Euro 2016 qualifier away to Latvia.
They are third in Group A with seven points behind leaders the Czech Republic, who have 13, and Iceland on 12. The top two qualify for the finals, along with the best third-placed side.
Netherlands striker Klaas-Jan Huntelaar scored with two headers either side of a Gyasi Zardes equaliser before his thunderous shot for what would have been a hat-trick deflected in off Memphis Depay as the Dutch looked on their way to a comfortable home win.
Huntelaar had opened the scoring in the 27th minute as he stretched his body amid a cluster of American defenders to steer home a cross from Manchester United-bound Depay.
A Fabian Johnson cross set up an American equaliser four minutes later. Zardes squeezed home a difficult right-foot effort as he came steaming in from the left wing.
Huntelaar restored the Dutch lead soon after half-time with a perfectly-timed header from Gregory van der Wiel’s cross and four minutes later, in the 53rd, he hit a powerful shot from just inside the area that caught Depay flush in the face and went in.
DeAndre Yedlin, who joined Tottenham Hotspur in January, proved a revelation for the U.S. after coming on and played the decisive pass for John Brooks to make it 3-2 in the 71st.
Yedlin and Bradley then carved open the Dutch defence on several occasions, as the Americans took control of the later stages and engineered their first triumph over the Netherlands.
Watch the Netherlands’ European Qualifier away to Latvia in Group A live on Sky Sports 5 HD next Friday, via the Red Button, from 7.45pm.