"It is a dream for a boy like me being here and to play for my home club is unreal," Chalobah said after the game.
"As I was getting closer to the goal I though I might as well shoot and just went for it.
"I didn't know the ball went in, I did not know what to do. I was over the moon.
"I have been training with the team in preseason and the manager said I was playing and I was ready. I was surprised by the Super Cup game too and I have taken the opportunity with both hands."
Alonso put Chelsea ahead on 27 minutes with a free kick. He swept the ball over the Crystal Palace defence with his left foot.
United State men's national team international Pulisic added Chelsea's second 10 minutes later when Vicente Guaita saved a Mason Mount shot. The goalkeeper only managed to push the ball as far as Pulisic, though, who tapped it into the net.
Chelsea's third came from Chalobah -- who used to be a ballboy at Stamford Bridge -- on 58 minutes. He struck the ball into the bottom corner from 25 yards out.
Tuchel, who took over Chelsea midway through last season and led them to a top-four finish, said his team are "fourth favourites" to win the league and must hunt down Manchester City, Manchester United and Liverpool - last season's top three teams.
"To be totally honest, I see us as the number four, that cannot be the favourite in the next year," he told reporters after Saturday's game.
"This is the Premier League, a tough competition. It's on us, we are in this role and in the role of the hunters, we have to close the gap to Liverpool, Manchester City and Manchester United.
"This was a very good first step, but that's the reality and we are not shy to say this out loud that we try to be the best in any competition we play, but the reality is we come from fourth place and have three teams to close the gap to."
The result means Patrick Vieira starts his tenure at Crystal Palace with a loss. The ex-New York City FC boss was appointed coach after Roy Hodgson stepped down during the summer following four years in the role.