The 28-year-old, who resumed on 89, never found the same fluency that had characterised his innings on Friday.
It was no shock when Boult, who had removed all of England's top three with a superb display of new-ball bowling, located Overton's outside edge.
Surrey paceman Overton received a rousing reception as he returned to the pavilion and the cheers kept coming.
England's new leadership duo of captain Ben Stokes and coach Brendon McCullum had urged the team to play with freedom after taking charge of a side that had won just one out of 17 Tests prior to this series.
Stuart Broad, a fast bowler by trade but also a fine shot-maker earlier in his Test career, embraced the new approach by hitting Boult for 14 runs in three balls courtesy of two fours and a six to the delight of the crowd.
Broad was eventually bowled by Tim Southee for 42 but by then England had gone ahead.
Spectators had already had plenty to cheer as Yorkshire favourite Bairstow, fresh from his match and series-winning 136 against the World Test champions at Trent Bridge, went to 150.
But one ball after Broad's exit, he holed out off spinner Michael Bracewell to end a brilliant 157-ball innings including 23 fours.
Number 11 Jack Leach hit two fours before he was last man out, lbw to Southee.
Bairstow then resumed his former role of wicketkeeper, with Ben Foakes unable to take the field because of a stiff back.
Stokes took the unusual decision to give a spinner in Leach the new ball following the left-armer's 5-100 in the Black Caps' first innings.
New Zealand, however, got through five overs before lunch, without alarm.