Finn's 199-ball 56 not out was a dour but increasingly significant contribution once wickets began to fall at the other end in England's teatime 382 for four.
They therefore had a lead of 89, though - and with only 24 overs remaining on the final day, a drawn first Test was highly likely.
Compton's was the only departure of a sunny morning, to Neil Wagner (three for 124). But Jonathan Trott (52) and Kevin Pietersen also went to the left-arm seamer as the onus fell increasingly on Finn to keep the Kiwis at bay.
Compton (117) augmented his maiden hundred by just 15 runs before falling lbw; Trott shared a third-wicket stand of 90 with Finn but went caught-and-bowled, and then Wagner got Pietersen for the second time in the match following his first-innings golden duck. Finn, meanwhile, not only improved his career-best twice in the same Test but dug in for a cussed maiden half-century.
Compton reopened his account with a trademark extra-cover drive for four off Tim Southee, but was to be the only casualty before lunch - falling over slightly in defence as Wagner found just a hint of inswing. His 310-ball vigil was therefore over after almost seven hours at the crease, mostly in company with captain Alastair Cook in an opening stand of 231 yesterday. Thanks almost entirely to them, England already had obvious prospects of ensuring the draw on a pitch showing no signs of deterioration appeared obvious.
Finn, who had come to the crease last night and accompanied Compton to his hundred once Cook went, batted with increasing assurance too. He needed just one minor moment of fortune this morning 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.
Trott was fluent on his way to a 78-ball 50, which he completed with his eighth four - a trademark shot past straight midwicket off Southee. But the return of Wagner undid him, extra bounce inducing a return catch in startled back-foot defence, and Pietersen never looked comfortable in his short stay before he got a faint inside-edge behind.
Finn's stoic resistance verged on the strokeless after he reached his 50, as he came to terms with his new responsibilities, and he went 49 balls without a run once Trott was gone.
But England were surely grateful nonetheless.