First change Finn took two wickets for one run in 16 balls to remove New Zealand openers Peter Fulton and Hamish Rutherford.
At lunch, New Zealand were 292 runs behind England's first innings 354. Kane Williamson was four not out and Ross Taylor unbeaten on nought.
New Zealand had been shot out for just 68 in their second innings during a crushing 170-run defeat in the first match of this two-Test series at Lord's last week.
Then a superb over to Rutherford saw Finn beat the left-hander three times in a row outside off-stump before he had the batsman edging to Ian Bell in the gully for 27.
England, 337 for seven overnight after Joe Root (104) became the first Yorkshireman to score his maiden Test hundred on his Headingley home ground, lost their final three wickets in today's first five overs.
All of his wickets came within a second new ball, starting Saturday, of five for 24 in seven overs.
Tim Southee, who took 10 wickets at Lord's, struck first today when he had had Matt Prior well caught by first slip Taylor after the wicket-keeper had added just one to his overnight 38.
Boult then knocked back Finn's off-stump before holding a sharp caught and bowled chance to dismiss last man James Anderson.
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