Djokovic was makings his first appearance since being stunned by Hyeon Chung at the Australian Open.
In their only other ATP meeting in 2013, Delbonis won on clay in Hamburg 7-6 (7), 7-6 (4). Daniel plays next Argentinean tennis player Leonardo Mayer.
Djokovic said it felt like the "first match I ever played on the tour".
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Kasatkina will play either seventh seed Caroline Garcia of France or Angelique Kerber of Germany in the quarter-finals. Earlier on Tuesday, Halep booked a quarter-final place against Petra Martic after getting the better of China's Wang.
Nadal isn't playing at Indian Wells or the Miami Open after aggravating the leg injury that forced him out of the Australian Open in January.
"Very weird. I completely lost rhythm", admitted the Serb.
"So I'm on a good run right now, and I try to maintain that, and you only maintain that if you respect every opponent", the 20-time men's singles Grand Slam champion said.
Djokovic scrambled to win the second set on Sunday but the rust from the layoff was evident as he made 62 unforced errors overall.
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Frenchman Gael Monfils survived a match point before beating big-serving American John Isner 6-7 (5) 7-6 (3) 7-5 in a contest that had only one break.
Playing for a third consecutive day due to weather delays in Indian Wells, the 36-year-old Swiss showed no sign of wear en route a 6-2 6-1 thrashing of the Serbian. "Back in the day, I used to play six matches in a row".
On Saturday, Federer led 6-3, 2-2 when the match was delayed by a heavy drizzle 56 minutes after it began.
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The van's driver is in stable condition. "Praying that the injuries will heal and that they receive great care", Hacker said. The Sheriff's Office, with the assistance of the State Patrol, is conducting a crash reconstruction.
On his return from right elbow surgery, Djokovic looked a pale shadow of his former self as he went down 7-6 (3) 4-6 6-1 to make a quick exit from a tournament he has won five times. Fifth seed Karolina Pliskova too needed three sets to reach the fourth round but the French Open titlist Jelena Ostapenko bowed out.