Klopp brought on Brewster in the closing moments, with the game heading to penalties.
The young forward then hit the bar, to hand the Gunners an advantage in the shoot-out.
Brewster looked distressed after the match, but Klopp has insisted the player is not to blame.
“This will not harm Rhian. If anyone is to blame, it is me.
“I wanted him in the position because he is a real finisher and confident. Today, not. That happens in life and in football.
“We all have to learn and prove we can deal with defeats. If he would have scored and someone else missed, it is the same thing,” Klopp told BT Sport.
It is the second successive year Liverpool have lost the Community Shield on penalties.