Scotsman Murray overcame Taiwan's Lu Yen-Hsun 6-3 6-2 on the Stadium Court in an evening encounter after top seed Djokovic had beatedn Bulgaria's Grigor Dimitrov 7-6 6-1 in an error-strewn match in hot afternoon conditions at Indian Wells.
Seventh seed Juan Martin Del Potro of Argentina brushed aside Germany's Bjoern Phau 6-2 7-5 while eighth-seeded Frenchman Jo-Wilfried Tsonga withstood a fast start by Mardy Fish to overcome the American 7-6 7-6.
World number three Murray was delighted to gain revenge against Lu, who beat the Scot in their only previous meeting, in the first round of the 2008 Beijing Olympics.
"It was a tough match," Murray said after breaking his opponent's serve twice in each set while also having to battle hard to hold his own serve early on.
"He started off well. I didn't serve particularly well to get myself into those situations, and then I served well to get myself out of some tough situations.
"He can kind of leave you flat-footed sometimes, and he certainly did that a lot in the first set. I knew it was going to be a tough encounter, so it was good to win in straight sets."
US Open champion Murray will next face Carlos Berlocq of Argentina, who eased past Japan's Kei Nishikori 6-2 6-2.
Djokovic came from 2-5 down to win his opening set 7-4 on the tiebreak, then broke his opponent's serve in the second and sixth games of the second set to seal victory in just over an hour.
Dimitrov, who at 21 was the youngest player left in the draw, made four double faults when serving for the first set at 5-3 and his challenge then quickly folded.
Serb Djokovic extended his winning streak to 20 matches, his last defeat coming in October to big-serving American Sam Querrey in the second round of the ATP Masters 1000 event in Paris.
Intriguingly the pair will meet again in the last 16 at Indian Wells, Querrey having scraped past Australian Marinko Matosevic 7-6 6-7 7-5 earlier in the day.
"I tried to hang in there and stay in the match," a relieved Djokovic said after improving his record for the year to 15-0. "I started very bad, very poor.
"He served very well but he made four double faults at 5-3 and got me back into the match. He made a lot of unforced errors and I just had to get the ball back in the court."
Asked about the challenge of Querrey in the fourth round, Djokovic replied: "It will be very interesting. He won our last encounter in Paris a few months ago. He has a very big serve."
In other matches, Canadian Milos Raonic fired down 16 aces as he battled past Croatia's Marin Cilic 3-6 6-4 6-3 while Germany's Tommy Haas won a cliff-hanger against Spaniard Nicolas Almagro 6-3 6-7 7-6.
BNP Paribas Open results
3-Andy Murray (Britain) beat Lu Yen-Hsun (Taiwan) 6-3 6-2
7-Juan Martin Del Potro (Argentina) beat Bjoern Phau (Germany) 6-2 7-5
Carlos Berlocq (Argentina) beat 16-Kei Nishikori (Japan) 6-2 6-2
1-Novak Djokovic (Serbia) beat 31-Grigor Dimitrov (Bulgaria) 7-6(4) 6-1
19-Tommy Haas (Germany) beat 11-Nicolas Almagro (Spain) 6-3 6-7(2) 7-6(2)
23-Sam Querrey (U.S.) beat Marinko Matosevic (Australia) 7-6(5) 6-7(7) 7-5
8-Jo-Wilfried Tsonga (France) beat 32-Mardy Fish (U.S.) 7-6(4) 7-6(0)
17-Milos Raonic (Canada) beat 12-Marin Cilic (Croatia) 3-6 6-4 6-3