Puncheon, who spectacularly missed from the penalty spot in Palace's 2-0 defeat at Tottenham last weekend, made the most of a mistake from Stoke winger Oussama Assaidi in the 51st minute to fire low into the net and hand the hosts their first win in four league games.
Palace, bottom of the table on Saturday morning, climb out of the Premier League relegation zone and up to 16th as a result.
The win had additional significance for ex-Stoke manager Pulis, who spent 10 years at the Britannia Stadium over two spells before leaving at the end of last season, and the 56-year-old was given a fine reception from both sets of fans as he emerged from the tunnel for the start of the game.
Stoke are now winless in five Premier League games, leaving them a mere two points above Palace.
Both sides made one change from their previous outings. Palace brought in Adlene Guediora for Cameron Jerome, ineligible against his parent club, while Assaidi returned at the expense of Marko Arnautovic for Stoke.
In an attritional first half, clear opportunities were few and far between.
Joel Ward headed Yannick Bolasie's cross wide in the fifth minute, before Marc Wilson's downward header was brilliantly tipped round the post by Julian Speroni 10 minutes later.
Charlie Adam went close for the visitors in the opening minute of the second half, collecting Jon Walters' cushioned header and beating Damien Delaney before firing a right-footed shot a yard wide of the post from the edge of the area.
Five minutes later Palace moved in front as Puncheon capitalised on Assaidi's error. The Moroccan did well to track the run of Adrian Mariappa and regain possession, only to gift the ball straight to Puncheon, who made space for himself before firing low across Jack Butland and inside the left-hand upright.
Stoke midfielder Adam fizzed an effort narrowly wide from 30 yards in the 63rd minute, while substitute Stephen Ireland fired a first-time volley over the crossbar from a tight angle.
The hosts appeared content to sit back and hold onto their lead as Stoke struggled to create any clear goalscoring opportunites, with Peter Crouch curling a shot well wide of Speroni's goal from 20 yards.
Palace were only denied a second by some heroic goalkeeping from Butland six minutes from time, the England international first denying Ward, before jumping to his feet to block Puncheon's follow-up and somehow smothering substitute Stuart O'Keefe's rebound.
There was still time for Butland to deny Puncheon in stoppage time, but in the end it mattered little as the hosts held on to seal a crucial three points in their fight to avoid the drop.
