Playing her first match since taking over at the top, Spain's Wimbledon champion broke her Puerto Rican opponent twice in the first set and was even more dominant in the second.
Puig dropped serve in the first game of the match but used her impressive court coverage to restore parity in the fourth game before Muguruza seized control of the contest when she broke her opponent again in the next game.
After winning the first set, the 23-year-old Muguruza moved up a gear to race away from her 70th ranked opponent and sealed victory with her third ace after 75 minutes.
"It's a big change for me," top seed Muguruza said in a courtside interview when reminded she had entered the tournament five years ago as a qualifier.
"I improved so much and I think everybody can see it. I'm very happy about that," added the Spaniard, who will meet either home wildcard Kurumi Nara or Caroline Garcia of France in her fourth successive quarter-final appearance in the event.
Earlier, second seed Karolina Pliskova appeared to suffer no ill-effects of playing without a coach when the tall Czech swept aside Magda Linette 6-2 6-1 to also reach the last eight.
The world number four, who split with coach David Kotyza last week, capitalised on an early break to win the first set in little more than half an hour.
Pliskova's power proved too much for Linette, who was broken in the second game of the second set, before the Pole's wayward forehand helped her opponent book a place in the next round.
"I felt great today on the court," Pliskova said. "I think my serve was working well... I'm really happy with my win today."
She now faces seventh-seeded German Angelique Kerber who overcame Russian Daria Kasatkina 7-6(5) 6-3.
Another Russian Anastasia Pavlyuchenkova laboured to a 6-4 4-6 6-2 win over China's Wang Qiang, who inflicted a 'double bagel' on eighth seed Kristina Mladenovic in the first round.
(Reporting by Amlan Chakraborty in New Delhi; Editing by John O'Brien and Ken Ferris)
