Chelsea boss Frank Lampard is under pressure after his team slipped down to eighth position in the table with four defeats from their past six Premier League matches.
Chelsea have also failed to beat any of the Big Six teams this season, and Pulisic said the club need to start picking up positive results quickly.
"We're in a tough period," he told the club's website. "It's going to happen to everyone at times, and we've been in these situations ourselves before.
"We have a game next week where we need to start turning it around because we do need to start getting results quickly.
"We'll watch this game back, look at it as a team and see what we can improve on. It's not going to be easy, but we need to show character now.
"We have some games coming up where we need to turn it around quickly and start putting together a run of results."
Callum Hudson-Odoi scored a consolation goal for Chelsea, but Pulisic said the team needed to defend better to prevent City's goals.
"We conceded goals with a lot of counter situations, and we weren't prepared to stop that, so the first half really let us down," he said.
"They were able to beat us on some transitions where we needed to defend better and be a little smarter, and then we allowed them to dictate the tempo of the game."
Chelsea welcome League Two side Morecambe to Stamford Bridge in the FA Cup on Sunday before they face Fulham in a London derby next week.