Burke will be re-evaluated on Monday at which point it will be determined if he needs surgery.
Burke injured his finger in the first quarter, but returned to play nearly eight minutes in the second quarter before leaving the game for good. Burke had two points and four assists in the game.
"We'll see what the specialists say but it's definitely a downer for us," Utah coach Tyrone Corbin said. "This is another obstacle we'll have to overcome, but we'll put our heads together, see what we have and go from there."
Richard Jefferson had 13 points and eight rebounds to lead the Jazz, and Alec Burks added 12 points.
Los Angeles, which led 50-31 at the half and 78-53 after three quarters, finished at 55 percent shooting from the field.
The Jazz, who opened the preseason with a rout of Golden State but have been blow out two straight since, made only 36 percent of their shots.
The Clippers (2-0) repeatedly broke down Utah's defense, with Chris Paul getting 10 assists in 23 minutes and Darren Collison adding 10 off the bench. Los Angeles dominated the paint as well, outscoring Utah 42-26 in close.
Los Angeles also had 18 fast-break points to only two for Utah.
The Jazz next host Portland on Wednesday.