Summer signing Puncheon, who will spend the season at Selhurst Park after joining on loan from Southampton, was lively throughout on his second outing for the Eagles.
He helped to keep the Sunderland defence on their toes and played a leading role in a 3-1 win which saw Ian Holloway's side open their points account for the season.
Palace left it late to secure the spoils, after Steven Fletcher had cancelled out an early opener, but Puncheon believes they were worthy winners.
He told Sky Sports: "I think it was important for us to get a win today. Considering what happened last week (in a 2-1 defeat at Stoke City), the most important thing was to get three points on the board, which we've done.
"I think the important thing today was when it went 1-1, when they got their goal, to come from drawing to win the game - which is important in the Premier League.
"It was a hard decision for me coming from Southampton, I enjoyed my time there, but the most important thing for any footballer is to play football. That's what I'm doing now, so I'm glad for that."
Palace's record signing Gayle started to repay the sizeable investment the club have made in him by successfully converting a second half penalty, with the 22-year-old holding his nerve to net his first Premier League goal.
He said: "I was delighted. It was a bit of a mishit, but it counts. I'm delighted to get the first one. I was very nervous, but it was a great team performance.
"They were giving me a bit of as stick, but luckily I got the goal and we got a third just to finish it off."