Bairstow is painfully close to a fifth Test ton, employing aggression against Neil Wagner and Ish Sodhi in particular in a 154-ball knock which has comprised 11 fours and a six.
James Anderson and Stuart Broad reduced the hosts to 36-5 in reply.
New Zealand lead the two match series 1-0.
Day three will start at 23:30 BST on Saturday.
They added 142 runs for the sixth wicket to help New Zealand cut England's lead to 115.
Left-arm seamer Trent Boult, who took four wickets in the first innings, has another two in the second innings while new-ball partner Tim Southee, who finished with six in the first dig, has the other scalp.
Cook's dismissal Sunday was demonstrative of his current lack of form and confidence as he unnecessarily played at a wide delivery from Boult and edged the ball to wicketkeeper BJ Watling.
However, Tim Southee (6-62) and Trent Boult (4-87) struck in successive overs to Leach and Bairstow respectively as England were bowled out for 307 in their first innings. In that game, two debutants in Jeet Raval and Colin de Grandhomme returned with sweet memories as de Grandhomme tore apart the Pakistan batting lineup in the first innings taking 6-41. The ball moved away a fraction and Cook followed it.
File image of England captain Joe Root and New Zealand captain Kane Williamson. A Test match is unfolding overseas that England are now firm favourites to win.
The tourists accelerated to 8-290 by stumps after being sent in, boosted by some lusty late hitting from Bairstow (97 not out) and Mark Wood (52).
"We talked as a group about wanting to attack the knee roll and get the brand new ball a touch fuller because we know it flattens out".
He said: 'Vince and Stoneman had a really good partnership, and we couldn't quite break through there. But we also used some different tactics. Losing only one test match out of four previously played at Hagley. "If you set the field well you can restrict the economy rate and create chances".
"That's the hardest I've had to bat at this stage of my Test career".
"It's a massive session [today]".
"I didn't want to work with any coaches - I just wanted to get that feeling back as a cricketer, and I've got that".
After putting on 70 runs in the first session, England slightly upped the tempo to 80 in the second.
"It didn't surprise me that Stokes caused problems when he came on - because he's injured, he can't just race in and bang it in". They began England's winter with Ashes half-centuries in Brisbane, and have ended it with two more much-needed contributions.