The former England captain's exit came early in the afternoon session, just moments after Ollie Pope was caught behind off Matt Henry for 18.
With Zak Crawley having been removed for a duck by Boult before lunch, England's hopes of making history with the highest successful run chase in a Trent Bridge Test were hanging in the balance.
Alex Lees and Jonny Bairstow were at the crease as England tried to clinch the three-Test series with a match still to play.
Test world champions New Zealand needed seven more wickets to level the series, with their mission complicated by the back injury that has deprived them of the services of pace bowler Kyle Jamieson.
Earlier on Tuesday, New Zealand were bowled out for 284 in their second innings to set England a challenging target on a flat Trent Bridge pitch.