Djokovic, the top seed in the elite ATP Masters 1000 event, was broken twice in his opening set against Spaniard Albert Ramos-Vinolas before advancing 7-5 6-3 in an evening match lasting just under an hour and a half.
"I managed to play the right shot at the right time in the important moments," said Djokovic, who is bidding for a fourth title at Indian Wells. "And I managed to sneak through in straight sets."
Scottish world number four Murray recovered from a surprising second-set lapse during the searing heat of the afternoon at the Indian Wells Tennis Garden to beat German Phillipp Kohlschreiber 6-1 3-6 6-1.
However, Spanish eighth seed David Ferrer made an early exit, ousted 7-5 6-4 by Australian Bernard Tomic in the third round.
Djokovic won a topsy-turvy opening set that featured five service breaks. The Serb failed to hold in the first game before breaking back in the sixth and eighth, but was surprisingly broken again in the ninth when serving for the set at 5-3.
Though Ramos-Vinolas fought back to level at 5-5, Djokovic held to love before breaking the Spanish left-hander again in the 12th, the set ending after 49 minutes when his opponent netted a backhand.
The second set went with serve until a protracted eighth game which the Serb won when Ramos-Vinolas blasted a low forehand wide, then served out to love in the ninth to seal victory.
BIG-SERVING OPPONENT
Djokovic will next face big-serving American John Isner, who powered past South African Kevin Anderson 7-6(6) 6-2.
"He's one of the best servers in the history of the game," said the Serb. "I expect a very tough match."
Murray appeared in total control after breezing through the opening set in just 25 minutes, but was broken after a marathon eighth game in the second as Kohlschreiber levelled the contest.
It was then back to business in the third for Murray as he broke the German's serve in the second and fourth games before sealing victory in a little under two hours when his opponent hit a forehand wide.
"Towards the end of the second set there were a lot of long games and he was creating a lot of opportunities," Murray said.
"The game where I got broken, I played a few poor shots as well. He capitalised on that. But apart from that, I thought I played a good match against a tough opponent who plays well in these conditions."
In other matches, fifth-seeded Japanese Kei Nishikori fought back to beat Spanish left-hander Fernando Verdasco 6-7(6) 6-1 6-4 while Australian Thanasi Kokkinakis scraped past Juan Monaco of Argentina 6-2 5-7 7-6(5).
(Editing by Steve Keating/Peter Rutherford)
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