The Potters were awarded the spot kick after Ashkan Dejagah handled in the area but Cottagers keeper Mark Schwarzer made the stop to deny Walters.
Pulis told the club's official website: "It was the decisive point. We've played a lot away games this year that have been very tight games, and again today it was a very close and tight game.
"When you get a chance like that you have to score them. If you get the goal, it gives the team more confidence to push on."
Following their defeat, Stoke are now 10th in the Premier League.