The former Stoke boss has dragged the Eagles from the bottom of the Premier League to 12th in the table following a third successive victory over Aston Villa on Saturday.
Palace, newly promoted from the Championship, lost nine of their opening 12 games under previous boss Ian Holloway but have picked up 10 wins since Pulis took charge in November.
And Puncheon, who scored Palace's winner at Selhurst Park, said: "I think he will definitely be up there with a chance of winning manager of the season.
"If you look at the position that we were in when he came and what he's done to this team - we all have the belief that we will stay in the Premier League now."
Puncheon, who signed from Southampton last summer, has scored six goals since the turn of the year but he is determined to stay grounded until safety is assured.
"We're not safe until it's mathematically impossible. We've just got to keep going until we reach that point," he added.
"It was a great win, just like the two we had before, but we don't get carried away. We've got to go to Everton and then West Ham after that, so we've got two tough away games in the next week."