Callum Wilson's second-half penalty looked like sealing victory for Newcastle who played virtually the whole game a man down after Ciaran Clark was sent off in the ninth minute for inexplicably hauling back Pukki.
Wilson converted in the 61st minute after a header hit Norwich midfielder Billy Gilmour's arm.
Norwich had failed to make the most of their numerical supremacy but Pukki struck a sensational volley to earn Dean Smith's side a point in the 79th minute.
The visitors should have claimed victory late on but Pierre Lees-Melou failed to take a glorious opportunity.
Newcastle are now without a win in their first 14 league matches and stay bottom with seven points while Norwich, who are unbeaten in four games, moved up to 18th spot with 10 points.
Newcastle captain for the night Wilson could not hide his frustration after the game but refused to criticise Clark.
"We knew how big the game was. At 11-v-11 I think we'd win," said Wilson. "Mistakes happen in football. We're not going to dig anyone out.
"We kept the door shut until the end but it's a great finish. We can't feel sorry for ourselves."
Before kickoff Norwich might have been happy with a point, but they were left rueing an opportunity to put daylight between themselves and Newcastle with a third win in four.
"We're frustrated. I'd have preferred Teemu to go straight through and score the goal," Smith, who is unbeaten in three games as Norwich manager, said.
"We had to change our game plan a little bit and they were always going to try and kill the game at every opportunity.
"They got a dubious penalty which lifted the crowd and gave them something to hold onto."
Newcastle had started the game well with Jonjo Shelvey firing over but everything changed when Clark cleared a ball straight at Pukki who was suddenly through on goal.
Pukki still had plenty to do but Clark blatantly hauled him back and referee Andy Madley no option but to show the red card.
Norwich looked nervous though and fell behind when Gilmour inadvertently blocked Federico Fernandez's header with his arm and, after a lengthy VAR check and a visit to the pitch-side monitor, Madley awarded the penalty.
Wilson's kick was not the best and former Newcastle keeper Tim Krul almost kept it out, the ball going over the line off the underside of the crossbar.
Norwich struggled to put any pressure on Newcastle's defence but levelled when substitute Dimitris Giannoulis's first-time cross was met with a spectacular volley by Pukki.