The beleaguered Magpies, who had defender Ciaran Clark sent off after just nine minutes, looked to be on course to end their drought when Callum Wilson converted a 61st-minute penalty awarded for handball after a VAR review.
However, Pukki’s sumptuous 79th-minute volley robbed head coach Eddie Howe, belatedly in the home dugout for the first time, the victory he so desperately needed and left the Magpies rooted to the foot of the table.
With skipper Jamaal Lascelles and wing-back Matt Ritchie suspended, Howe abandoned the five-man defence with which he had started his first two games in favour of a 4-2-3-1 formation with Joe Willock partnering Jonjo Shelvey in the middle of the field.
However, his game-plan was ripped up within nine minutes when, having miskicked his attempted clearance straight at Pukki, Clark dragged the striker back and was shown a straight red card by referee Andy Madley.
Goalkeeper Martin Dubravka spared the home side further pain with a good save from Billy Gilmour as he took charge of the resulting free-kick as Howe sacrificed winger Ryan Fraser to send on defender Federico Fernandez.
Nevertheless, it was the home side that almost took the lead with 20 minutes gone when Allan Saint-Maximin turned Gilmour and raced towards goal before squaring for Joelinton, whose rising drive flew just too high.
Norwich, however, were making good use of their numerical advantage to pin the Magpies back with Gilmour prominent, and it took a superb block by Canaries old boy Jamal Lewis to deny Josh Sargent after he had met Christos Tzolis’ 29th-minute cross at the far post.
Dubravka had to get down smartly to save Tzolis’ effort at his near post in first-half stoppage time, but there was to be no breakthrough before the whistle sounded.
Norwich returned with real purpose to force their hosts back once again, although they continued to lack the cutting edge to make the most of the pressure they were exerting.
For their part, the Magpies still looked dangerous on the break and Joelinton was aggrieved not to earn a 53rd-minute penalty after going down under Brandon Williams’ challenge.
However, Madley did point to the spot after being advised to take a second look at Fernandez’s 59th-minute header, which appeared to be blocked illegally by Gilmour, and although former Magpie Tim Krul got a firm hand to Wilson’s spot-kick, he could not keep it out of his net.
But Newcastle were pegged back when, after Dubravka had spilled Gilmour’s deep cross, Pukki volleyed home emphatically after Dimitris Giannoulis had recycled beyond the far post, although the Slovakia keeper redeemed himself at the death with a crucial stoppage-time bock from substitute Pierre Lees-Melou.