The Jazz led for much of the game after breaking a 15-15 with a 14-1 run late in the first quarter but a strong final period earned the Clippers a third straight win.
Blake put his side ahead in front for the first time since 2-0 when he dunked in a feed from Chris Paul with five minutes remaining in the game.
Utah responded with back-to-back baskets for Al Jefferson, who ended the night with 16 points and 10 rebounds, but the teams remained locked at 99 apiece in the final minute.
Paul netted a lay-up with 40 seconds to go and a pair of free-throws took Jamal Crawford's tally for the night to 20 and the Clippers' lead out to 105-101.
Randy Foye cut the deficit to one with a three-pointer then picked off Chauncey Billups' inbound pass but his desperate long-range effort was after the buzzer and off target anyway.