The midfielder was dismissed with 20 minutes remaining at the Rose Bowl Stadium for a late challenge on the Reds’ new signing Ragnar Klavan.
Gary Cahill got the only goal of the game in the 10th minute, heading in a Fabregas corner.
The England defender was also booked in the clash of the Premier League rivals, with Liverpool’s Alberto Moreno, Ovie Ejaria, Marko Grujic, Dejan Lovren and Kevin Stewart also receiving yellow cards.
As well as Kavlan, 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.
The closest his team came to a goal was in the 35th minute, when Roberto Firmino’s effort was ruled out for offside.