After trailing 3-5 in the opening set, Nadal produced a series of stunning returns to break Karlovic in the 10th game to claim the lead.
The Croat, however, refused to give up, prevailing in the second set tie-break to force the decider, which again went to the wire before the Spaniard claimed victory.
There was no such drama for Djokovic as the world number one dropped only three games against Klizan.
Eyeing a third Beijing-Shanghai double, Djokovic converted half of the 10 break points that came his way to sweep aside his error-prone Slovak opponent.
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"Klizan is a very flashy player," Djokovic, who won last week's China Open in Beijing, said.
"He can play very fast but with a tendency to make a lot of unforced errors, which he did. He made my life a bit easier on the court today.
"I was trying to serve, move him around the court. Just played as much as I needed. It was enough to win," the Serbian said after improving to a 13-match winning streak.
French Open champion Stan Wawrinka also maintained his winning run with a 7-6(3) 6-3 victory over Viktor Troicki to set up a third-round contest with Marin Cilic.
Third seed Andy Murray beat American Steve Johnson 6-2 6-4, Bernard Tomic eliminated seventh seed David Ferrer 6-4 6-2 and Japan's Kei Nishikori overcame Australian Nick Kyrgios 1-6 6-4 6-4.
(Reporting by Amlan Chakraborty in New Delhi; editing by Toby Davis)
