Bairstow and Overton, the fast bowler on debut, came together at 55-6 in reply to New Zealand’s 329.
The pair lashed out in a record unbeaten stand of 209 runs to lead their side to 264-6 by stumps.
Bairstow, playing on his home ground, was 130 not out from 126 balls. Overton was 89 not out from 106 on the second day of his maiden test.
They combined for 33 boundaries and two Overton sixes at a staggering run rate of 5.6 per over in England’s highest seventh-wicket stand to take the edge from New Zealand and put the test back somewhat on par.
Just hours after England looked likely to follow on, it was only 65 runs in arrears on the first innings.
England has the series sewn up after edging the first two tests, and the third is following a similar back-and-forth script.
New Zealand, 123-5 on Thursday, was all out just after lunch on Friday following Daryl Mitchell’s third century in three tests.
Mitchell was out for 109 on the stroke of lunch. Counting his knocks of 108 at Lord’s and 190 at Trent Bridge, Mitchell broke Martin Donnelly’s New Zealand record of 492 runs in a series in England, in 1949.
Boult clean bowled openers Alex Lees (4) and Zak Crawley (6), followed by Ollie Pope (5). Southee claimed Joe Root on 5. England was 21-4 — the top four dismissed in single figures at home for the first time since the 2005 Ashes test at Lord’s.
Neil Wagner added two wickets in a single over, taking out England captain Ben Stokes after a 13-ball cameo for 18 runs, and Ben Foakes lbw for a duck.
That reduced England to 55-6, and brought together Bairstow and Overton in the afternoon session.
Bairstow turned the second test with a remarkable last-day 136 in Nottingham last week, and merely picked up where he left off.
He passed 5,000 test runs driving Southee through the covers to the boundary. Consecutive boundaries off Southee brought up his fifty. Another boundary off Southee down the ground avoided the follow on.
Overton was no bystander.
A boundary off Boult to long off brought up their fifty partnership. His first six over long off, at spinner Michael Bracewell’s expense, brought up their century stand.
Overton, who has one first-class century to his name, raised his test fifty hitting Wagner through midwicket for his seventh boundary. He hit Wagner for his second six high over the midwicket fence in the 60s.
Bairstow reached his ninth test century, off 95 balls, smacking Boult past mid-off with his 15th boundary.
His 20th boundary brought up the 200-run partnership with Overton.
By stumps, Bairstow had 21 fours, Overton had a dozen, and the bowling figures of the New Zealanders had blown out.