The Blues ran out 2-0 winners against their London rivals at Stamford Bridge on Saturday afternoon, moving ahead of Manchester City and into second place in the process.
Neither side could find an opening goal in the first half, but Tammy Abraham broke the deadlock shortly after the interval, producing a neat finish after being played through by Willian.
Pulisic's 79th-minute header ensured Chelsea picked up another important three points just before the international break, with a huge clash against City up next on November 23.
The United States star claims the Blues will now try to chase down Premier League leaders Liverpool after moving to within five points of the summit.
Pulisic told reporters post-match: "I just got in the box and luckily the ball just bounced right to me and that helped us a lot, but I think we deserved it in the end. We had a lot of chances and we should've scored [more] but it was a good result in the end.
"The first half, we were dominating but we just couldn’t get the goal. We just stuck with it and eventually Tammy got the other one and in the end, it was a good result.
"Not bad at all [being second in the Premier League]. We want to be right up there at the top so we’ll just keep going."
Abraham notched his tenth goal of the season against the Eagles, which puts him level with Leicester City's Jamie Vardy in the race for the Golden Boot.
The 22-year-old is proud to keep defying critics who suggested he wouldn't be able to shoulder Chelsea's main goalscoring burden after returning to the Bridge from a loan spell at Aston Villa this summer.
"It wasn’t our best game today, you can see that", said Abraham. "The most important thing was that we got the three points. At half-time the manager said we need to stay focused as the chances will come and we came out in the second half much better and got the two goals.
"Crystal Palace are a good side. They defend well as a team and in the first half, they frustrated our style of play. Second half we believed and we didn’t change our style of play, we just wanted to play attacking football and we got the two goals.
"The No.9’s who have been at Chelsea have been world-class strikers and it was a big task for me to take on but I was ready, I felt ready in myself and I’m wearing it and I’m proud to be the No.9."