Vettori produced his heroics after the BLACKCAPS had declared on 262-6 after being asked to bat in a match which saw the first three days washed out at the Shere Bangla National Stadium.
The left-arm spinner left the hosts reeling at 13-3 before bad light denied him the chance of wreaking more havoc with the hosts requiring 163 to avoid the follow-on.
Entering the attack in the 10th over, Vettori grabbed a wicket with his first ball after Zunaed Siddique suffered what could only be described as a brain explosion. He charged down the track only to miss the ball and be stumped by Brendon McCullum at the second attempt after initially dropping the ball.
Bangladesh captain Mohammad Ashraful lasted only two balls before being trapped in front and with his last ball of the over he repeated the dose to remove Rajin Saleh.
He finished with the stunning figures of 3-0 from his solitary over.
Earlier Jesse Ryder came agonisingly close to a century in his second test but fell nine runs short.
Ryder batted responsibly to build his innings, accelerating once he passed fifty for the first time in test cricket. Three figures appeared on the cards as he clubbed 12 boundaries in his 140-ball stay, but got a top edge to Abdur Razzak while trying to paddle a ball to the leg side to offer Mehrab Hossain the simplest of catches.
The burly left-hander shared in a face-saving 137-run partnership with McCullum (66) after the BLACKCAPS had slumped to 49-3.
The pitch, which had been under covers for four days, had a glossy sheen to it and the BLACKCAPS openers were soon sloping off back to the pavilion.
Aaron Redmond shouldered arms to a ball that nipped back off the seam from Mushrafe Mortaza to be trapped in front and Jamie How followed two balls later when he was beaten by an outswinger from debutant Mahbubul Alam which crashed into his off stump. At 10-2 the tourists were in danger of handing Bangladesh the upper hand but Ross Taylor helped steady things with Ryder.
But when he fell for 19, leaving a ball from Shahadat Hossain that jagged back to hit the top of his off stump, and Ryder and McCullum had to start a salvage act.
Ryder went shortly before tea and after the break McCullum looked to up the ante but mistimed a slog-sweep to hole out to Ashraful off Shakib Al Hasan. The wicketkeeper batsman seemed to be close to rediscovering his best form after a wretched time of late with the bat.
Vettori hit a breezy 22 before getting an inside edge to a ball from Ashraful and it was left to Daniel Flynn and Grant Elliott to take the BLACKCAPS top a point where Vettori was happen to declare. Flynn managed a ponderous 35 from 81 balls after being dropped on 10 and Elliott was unbeaten on 8.
The BLACKCAPS lead the two-match series 1-0.
The tourists made one change with Grant Elliott replacing the injured Jacob Oram while Bangladesh handed a test debut to paceman Mahbubul Alam who took the place of Naeem Islam.