The Serbian had a little hiccup when he dropped the second set of his match with Colombia's Alejandro Gonzalez but was otherwise untroubled in winning 6-1 3-6 6-1.
"I thought I played really well from the start," said Djokovic, a two-times champion at Indian Wells.
"Then suddenly I just had a big loss of concentration and allowed him to win the second set for no reason. I bounced back better in the third."
While he was not at his best, Djokovic did at least survive to fight another day in the Californian desert after the shock losses of Rafa Nadal and Maria Sharapova on Monday.
"I obviously cannot allow myself to have these particular concentration lapses in the match at this level especially in the next match when I'm playing Cilic, a guy who is in really good form and I think has gotten better in last couple of months working with (Goran) Ivanisevic."
While Robredo lost to the towering Cilic, three of his compatriots all advanced with Fernando Verdasco upsetting world number nine Richard Gasquet of France 7-6 (7-5) 6-1 to leave Djokovic as the only top-10 player left in the bottom half of the men's draw.
Roberto Bautista Agut held his nerve to beat Finland's Jarkko Nieminen 6-2 4-6 7-6 (8-6) to move on before Feliciano Lopez joined the Spanish Armada with a 1-6 6-3 6-4 defeat of Mikhail Kukushkin of Kazakhstan.
BNP Paribas Open results
12-John Isner (U.S.) beat Lu Yen-Hsun (Taiwan) 6-4 6-3
Julien Benneteau (France) beat Dominic Thiem (Austria) 7-6(4) 6-3
20-Ernests Gulbis (Latvia) beat 15-Grigor Dimitrov (Bulgaria) 2-6 6-1 7-5
2-Novak Djokovic (Serbia) beat Alejandro Gonzalez (Colombia) 6-1 3-6 6-1
Feliciano Lopez (Spain) beat Mikhail Kukushkin (Kazakhstan) 1-6 6-3 6-4
30-Fernando Verdasco (Spain) beat 8-Richard Gasquet (France) 7-6(5) 6-1
Roberto Bautista (Spain) beat Jarkko Nieminen (Finland) 6-2 4-6 7-6(6)
24-Marin Cilic (Croatia) beat 16-Tommy Robredo (Spain) 6-4 6-3
- Sports & Recreation
- Marin Cilic
- Tommy Robredo