The Potters trailed 1-0 at half-time but struck twice in the second half to claim all three points and their first win of the season at the second attempt.
Adam scored the equaliser and was happy to get off the mark for the campaign.
"It was nice to be in there (the box to score)," he told Sky Sports. "It's good to get me off the mark and hopefully that's one to the tally that we can get to at the end of the season. It's a great win for us in the first home game of the season."
The Scotland international admitted that Stoke had been second best in the first half but felt the way they reacted after the break ensured they were worthy winners.
"Yeah, first half we were disappointed the way we played," Adam commented. "We were never at the races and fair play to Crystal Palace, they came here, they dug in deep and they were the better side first half.
"We showed a bit of character second half and thoroughly deserved the game, I think, in the end."
The victory was the first under new manager Mark Hughes in his first home match but Adam was happier to focus on the team's triumph and how they turned the game around.
"When you have a new manager there's a lot of hype in the stadium, the first home game and all that but for us, I think we played well in parts last week," he added.
"In the second half we got a reaction after half-time which was needed and overall I think we deserved three points."