Arsene Wenger’s side were kept at bay throughout the first half, but moved ahead two minutes after the restart through a lofted finish from Alex Oxlade-Chamberlain.
Cameron Jerome had a great chance to level almost immediately, but could only head straight at Wojciech Szczesny.
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Oxlade-Chamberlain then made Palace pay with a drilled effort on 73 minutes.
"We were disappointed to lose a poor goal after half-time," Pulis said.
"Then our big moment came 90 seconds later and we did not take it. That was the difference.
"We kept going; we showed good determination and resilience.
"We have played at Manchester City, Chelsea and Tottenham Hotspur since I have been here and we have been a threat in those games with our wide men playing really well.
"It was not the case today; they did not carry that threat, but they kept going."
Palace's main problem this season has been a lack of goals, with their return of 15 lower than any other side in the Premier League.
Pulis is hoping deadline-day signings Tom Ince and Joe Ledley, as well as Glenn Murray, who is set to return from a knee injury, can help provide a greater threat.
He added: "We need to score more goals and we're hoping our signings will do that. Tom Ince has goals from wide; we're convinced he will score goals. Joe Ledley has a good scoring record, so we're adding goals.
"Added to the fact that Glenn is nearing a return - he will play another game tomorrow [Monday] - we're feeling positive."