The 35-year-old needed just 62 minutes to dispense with his Chilean opponent 6-1 6-1, dropping only six points on his serve in the entire match.
Djokovic told the ATP Tour's official website: "From the start to the end, (it was) a great performance.
"Playing in a new tournament, different conditions, the first match is never easy. Obviously you are looking to see how you are going to adapt, but I did it perfectly, really, played as well as I can.
"I just made him work and used every opportunity that was presented to me."
Having arrived in Kazakhstan from Tel Aviv, where he collected his 89th Tour-level title last week, Djokovic was in imperious form once again as he climbed to 14th place in the Race to Turin, in which he needs a top-20 finish to make the end-of-season showpiece.
He will face world number 34 Botic van de Zandschulp in the next round.
Also advancing was third seed Stefanos Tsitsipas, after battling past up-and-coming Italian Luca Nardi 7-6 (7-2), 7-6 (7-3).
The Greek star did not find it easy to see off his young opponent, who made a nuisance of himself throughout the match.
"He's a player that can play very well in the future," said Tsitsipas. "I was impressed by his level, he was able to maintain that throughout the entire match. There weren't any holes. I believe today he was able to sustain that level from the beginning to end, and that was extremely impressive.
"I was trying to do my best, I was trying to switch it up as much as I can, but it wasn't as easy as I had it planned in my head. He made it difficult today."
Andrey Rublev enjoyed a more comfortable route to the quarter-finals as he eased to a 6-3, 6-2 victory over Zhang Zhizhen.