Breaks in the penultimate game of both sets proved the difference, Muguruza triumphing 7-5, 6-4 to book a meeting with Anastasija Sevastova.
Crawford, who first rose to prominence when she came through qualifying to reach the Brisbane semifinals 12 months ago, gave another good account of herself Down Under, overcoming a slow start to push the reigning French Open champion all the way.
Muguruza came out with her right thigh strapped after complaining of a sore leg during her first round, but showed few signs of sluggishness as she rattled off the first three games. Crawford began to settle into the contest and was soon back on level terms.
The pair exchanged a series of thundering baseline exchanges as the opening set reached its climax, Muguruza striking decisively with a backhand winner with Crawford serving at 5-5, 30-40. Mugurza went on to serve out the set. Muguruza held her nerve in an equally competitive second set to reach the last 32 for a fourth straight year.
"She was super-powerful and hit the ball very hard, which made it difficult," Muguruza said. "Today, I had to concentrate and wait for my opportunity."
"[My leg] is a little bit better. Honestly, I think the time is helping me. Hopefully now the next day is going to help me recover even more," Muguruza said. "I definitely had a concern. But I knew I had to go out there and play with what I have. You know, I went, try to play my normal tennis, try to be focused. It was important match. Tough opponent actually. I'm very happy with this win."