Bairstow (97 not out) found unexpected assistance from No 9 Wood (52) for a stand of 95 in 18 overs which switched the momentum of this second and final Test as England recovered to 290-8 at stumps.
Tim Southee (5-60) and Trent Boult (3-79), who had instigated England’s first-innings shambles of 58 all out last week, did all the damage as three wickets fell for one run in nine balls en route to 9-5, and later 164-7 in Christchurch.
By tea, it was already thanks only to Bairstow and Ben Stokes that England had salvaged a vaguely viable position.
What then followed from the eighth-wicket pair was refreshing in England’s long and hitherto miserable Test winter, with Bairstow on the brink of three figures by the close at a venue where he hit the fastest one-day international century by any English opener earlier this month.
It was perhaps a mild surprise that Kane Williamson put England in on a glorious morning, with the obvious caveat that memories of their Auckland capitulation were still very fresh in all minds.
Boult’s late movement with the new ball was too much for a leaden-footed Alastair Cook, whose off stump was spectacularly uprooted.
James Vince, back in at three to accommodate Joe Root’s return to his favoured No 4, rarely looked comfortable and third flirtation with the review system ended in an umpire’s-call lbw departure to Southee.
Root and Mark Stoneman closed out the session but got little further as England imploded after lunch.
Root had just driven Southee straight for four when he missed the next ball and was bowled, through the gate and off his back pad.
Boult got through a static Dawid Malan’s defences, lbw for a golden duck, and then Stoneman edged Southee to second slip. It was a routine end to an innings of laudable determination, plenty of transient fortune and ultimately little substance.
Bairstow and Stokes joined forces without a run between them and batted with the care of two men fully aware those still to come might struggle to add many more.
For the second time in succession, wickets soon began to fall on the resumption for a new session.
Boult broke the stand on 57 when Stokes flapped a catch behind down the leg side. Stuart Broad, pressed up to an unaccustomed No 8 for the first time in more than four years, presented Southee with his wicket via an easy catch to mid-off.
England looked in big trouble, but Wood and Bairstow had other ideas.
Wood followed solid defence with increasing readiness to put away anything loose in his maiden half-century, helping Bairstow pass 50 too when he dispatched successive short balls from Neil Wagner through the leg side for his sixth and seventh fours.
Bairstow was dropped by a leaping BJ Watling from an edge behind off Wagner on 58. Wood - playing his first Test since last July after a catalogue of ankle troubles - escaped half-chance tough catches on 28 and 48 off the left-armer.
No one could quibble at those, though, after a sparkling 50 from just 54 balls which contained seven fours and a memorable flat hook for six off Wagner.
He was eventually bowled trying to drive Southee, from the final delivery before the second new ball, but by then England had true reason to be thankful for Wood’s telling intervention.