Boult took 6-33 as Australia, who only needed to square the series to retain the trophy, were dismissed for 257 on the final ball of the 47th over as they chased New Zealand's 281 for nine.
Ben Stokes took two wickets in two balls to inspire England to victory over New Zealand in the first Test at Lord's.
New Zealand openers Peter Fulton and Hamish Rutherford safely negotiated their way to the close of play on the third day of the third Test against West Indies on Saturday after their bowlers had set up a likely victory with a destructive display.