Compton (117) augmented his maiden hundred by just 15 runs on a sunny final morning before he was lbw to Neil Wagner in the tourists' 329 for two.
That total put them 36 in front, with only 62 overs remaining in this rain-shortened first match of three. Finn batted throughout the session, not only improving his career-best twice in the same Test but putting himself within eight runs of a maiden half-century in an unbroken stand of 64 alongside Jonathan Trott.
Compton reopened his account in the fifth over of the day with a trademark extra-cover drive for four off Tim Southee.
He was to be the only casualty before lunch, though, falling over slightly in front-foot defence as left-armer Wagner found just a hint of inswing.
Compton's 310-ball vigil was therefore over after almost seven hours at the crease, mostly in company with his captain Alastair Cook in an opening stand of 231 yesterday.
Thanks almost entirely to them, with Finn and Trott doing their bit too, England's prospects of closing out a stalemate on a pitch showing no signs of deterioration were becoming ever more obvious.
Finn batted with increasing assurance, needing just one minor moment of fortune when he edged Kane Williamson's off-spin low and just out of the clutches of second slip for three to take him into the 40s in familiar surroundings following his stint with Otago in late 2011.
- Sports & Recreation