Substitute Jabo Ibehre scored the only goal of the game as Cambridge earned their first win of the season.
Ibehre broke the deadlock on 63 minutes when a shot from Uche Ikpeazu was parried by goalkeeper Glenn Morris and the striker tucked in the rebound from close range.
Crawley went close early on when midfielder Mark Randall rattled the crossbar with a dipping long-range effort.
Enzio Boldewijn failed to connect with a cross from Jordan Roberts for the hosts before Ikpeaku had a shot on the turn tipped over the bar by the alert Morris.
Crawley began the second half in lively fashion and Randall had a goal-bound shot blocked from the edge of the area but they fell behind through Ibehre's composed finish.
Dutch striker Thomas Verheydt and Matt Harrold both came close to equalising for Kewell's side in the narrow loss.
"I think you can see it's going to happen soon - we're getting closer," Kewell said.
"If things had gone our way then the result could have been so different today, we hit the woodwork twice and kept creating chances.
"Once the chances start to go in, you watch - there's a few tweaks we need to make but I believe our fortunes will turn around."
Kewell admits it hasn't been the best start to his first stint as head coach but has rallied his side to keep fighting.
"We've just got to keep working, keep fighting," he said.
"I mean I'm up for a challenge, I love a challenge and I want to have a challenge.
"No one said it was going to be easy, all these players know it's not going to be easy.
"But of we stick together and we fight, not only the players, the staff and the people who work here, but the fans as well, we'll get through it.
"But if we start seeing a divide, it's not going to help.
"I believe in every single one of these players that we can do it."