As well as Klavan, Liverpool boss Jurgen Klopp gave starts to fellow recent arrivals Sadio Mane and Loris Karius in a match which was watched by more than 53,000 in California.
Defender Cahill rose highest from a Cesc Fabregas corner in the 10th minute, heading it past Liverpool goalkeeper Loris Karius for the only goal of the game.
The closest his team came to a goal was in the 35th minute, when Roberto Firmino’s effort was ruled out for offside.
Liverpool showed some flashes of creativity and excitement around the box, but lacked a cutting edge up front, with Sadio Mane skilful but quiet, and showed far more muscle, as six of their players were booked.
Jurgen Klopp spoke to the media after the game. “First of all, I would say a little bit more freshness but that’s only with the finishing. For the last pass there were two things – a little bit fresher legs would help against a deep formation and the other thing is again more belief in our skills because we are able to play really good football.”
“We had a lot of good movements. I saw a lot of really good individual situations and team tactics, really good moments offensively and defensively. It was OK, we could have done better, but for the moment I am not dissatisfied. It is OK, not more.”
Liverpool recruit Marko Grujic was in the wars in the first period and had to be treated on the stroke of halftime after a clash of heads, and was replaced at the break by Adam Lallana.
The critics of Alberto Moreno will still be out as he had a nightmare 82 minutes. He was shown a yellow card early on for a rash challenge on Willian and generally used the ball poorly. Needs competition to push him further.
Chelsea sat back in the second half as coach Antonio Conte tinkered with his side, with Juan Cuadrado, Pedro and new signing Michy Batshuayi entering the fray.
In an interview with Chelsea TV, Antonio Conte said: “It was a hard match, but this game was important because we take confidence.
“I saw a lot of positive things about the defensive situation but we must improve when we have the ball because I love to play the ball, but we are improving.
“We must continue to work. Today we win. We showed a good attitude even when we went down to ten men.”