Eddie Howe's first game in the home dugout got off to a difficult start as the Magpies were reduced to ten men inside nine minutes following Ciaran Clark's red card after the centre-back's last-man challenge on Pukki.
Newcastle, however, broke the deadlock through Wilson's penalty towards the hour-mark after Federico Fernández's header struck Billy Gilmour's raised arm inside the penalty area but United were denied maximum points in the closing stages through Pukki's volley.
Howe made three changes to the side which suffered a 2-0 away defeat against Arsenal as Javier Manquillo, Clark and Jamal Lewis were all handed starts ahead of facing Dean Smith's Canaries.
The hosts created the opening goalscoring opportunity as Jonjo Shelvey, making his 250th Premier League appearance, controlled Allan Saint-Maximin's pass before rifling a strike from distance over the bar.
Despite an encouraging opening, Newcastle were then dealt a negative blow as Clark received a straight red card. After the defender's intended clearance fell into Pukki's path, the forward advanced clear of the Irishman before being pulled back, with Clark receiving his marching orders by referee Andy Madley.
Martin Dúbravka was on hand to deny Billy Gilmour's resulting free-kick, getting across quickly to vitally push the midfielder's effort way for a corner.
Despite their disadvantage, Newcastle continued their push for an opener as Joelinton almost netted a second goal in as many home matches. After Saint-Maximin completed a mazy run, the Frenchman teed up the Brazilian forward who drilled a first-time effort from the edge of the box but the ball flew narrowly over the crossbar.
Lewis, making his first league start of the 2021/22 campaign, produced a crucial block to prevent his former club from opening the scoring towards the half-hour mark.
As Christos Tzolis' cross from the left flank founds its way to Josh Sargent at the back post, the American's goal bound strike was importantly turned away by the left-back.
Wilson, captaining the Magpies for the first time, was unable to convert towards the closing stages of the first-half. Following Joe Willock's impressive run down the left flank, evading the challenge of two defenders, the former Arsenal midfielder picked out the striker inside the box who swivelled before firing his attempt over the bar.
In the second-half, United took the lead through Wilson's penalty. After Jonjo Shelvey's corner was curled into Federico Fernández's path, the substitute's header struck the arm of Gilmour and, after a thorough VAR check, Madley pointed towards the spot.
Despite former United goalkeeper Tim Krul guessing the correct way to Wilson's spot-kick, the Dutchman could only parry the attempt onto the underside of the crossbar before the ball bounced over the line to the delight of the home supporters.
Newcastle refused to hold onto their one-goal advantage as they pushed forward for a second goal, with Saint-Maximin slicing an effort wide of the far right-hand post after being played through by Wilson.
Isaac Hayden almost made an immediate impact from the bench as the central midfielder got onto the end of Wilson's pass from the byline before his sliding effort skidded across the six-yard area.
In the 79th minute, Newcastle found themselves on level terms through Pukki's equaliser. After Gilmour's initial cross into the box was clawed away by Dúbravka, the Magpies were unable to clear the loose ball before Dimitris Giannoulis' pass was lofted into Pukki's path before the Finnish forward's volley flew into the left-hand corner.
With full-time approaching, Dúbravka produced a crucial save in stoppage-time to deny Pierre Lees-Melou. After dispossessing Fabian Schär, the midfielder raced through on goal but his low effort was blocked by the right foot of the Slovakian shot-stopper before looping over the bar.
Newcastle United: Martin Dúbravka, Javier Manquillo, Fabian Schär, Ciaran Clark, Jamal Lewis, Jonjo Shelvey, Joe Willock (Isaac Hayden 72), Ryan Fraser (Federico Fernández 11), Allan Saint-Maximin (Miguel Almirón 77), Joelinton, Callum Wilson (c).
Subs not used: Karl Darlow, Emil Krafth, Jacob Murphy, Jeff Hendrick, Sean Longstaff, Dwight Gayle.
