Muguruza used her experience to get out of trouble, eventually set up a meeting with Ukrainian 10th seed Elina Svitolina in just over two hours under a relentless sun at Indian Wells.
“I started a bit off,” the 23-year-old Spaniard said. “My shots weren’t there. The experience of keeping calm and fighting helped; it’s no secret.
“I think I’m improving in general a lot of things, but I know that maybe two years ago this match I wouldn’t turn it around. I think that’s experience.”
Muguruza was impressed with Day, the U.S. Open junior singles champion.
“She reminds me a little of me, when I was super motivated and playing very good,” she said.
Day hoped to learn from her defeat.
“I was pretty upset after my match, because it didn’t go the way I wanted it in the end but if you look at the big picture, I was able to push a top 10 player really far,” she said.
On a day when three matches went to third set tiebreaks, there ultimately were no big upsets, with Czech third seed Karolina Pliskova and Russian eighth seed Svetlana Kuznetzkova also winning.
And fifth seed Slovakian Dominica Cibulkova also advanced, if only barely, saving a match point in a 2-6 7-6(5) 7-6(4) win over left-handed Czech Kristyna Pliskova, whose 18 aces were not enough.
Pliskova also had 13 break opportunities in the third set alone, but converted only two.
In an even closer encounter, Swiss 15th seed Timea Bacsinszky survived four match points to beat Dutch 18th seed Kiki Bertens 6-3 5-7 7-6(8).
Eleventh seed Johanna Konta was the highest ranked player to fall, the Briton edged 3-6 6-3 7-6(1) by Carolina Garcia of France.
(Reporting by Andrew Both in Cary, North Carolina)