After falling an early break behind, Suárez Navarro began to find her groove and in the end proved too strong for Van Uytvanck, running out a 64 60 winner.
"I think that for my first match it was a good one overall," Suárez Navarro said. "It took me a little time to adjust at the beginning and I believe that the first serve was key in the first set.
"In the second I played much better and more solid because I found my rhythm. I am pleased to have won this one today!"
The Spaniard is seeded No.3 this week and in the second round will face Alexandra Cadantu, who defeated Yanina Wickmayer, 36 61 61. Elsewhere there were also wins for Camila Giorgi, Annika Beck and Shahar Peer.
In Indian Wells, Giorgi gave a timely reminder of her precocious talent when she outgunned Maria Sharapova, and she took this momentum with her into her first outing back in Europe, racing past Katarzyna Piter, 60 61.
"It was my first match for a few weeks, so it has been a while. But I felt good out there," Giorgi, who will next play Ksenia Pervak or defending champion Roberta Vinci, said. "It was a good match and a good surface. There are still a lot of things I need to improve in my game but I´m pleased with my win."
Beck made a similarly quick start, winning the first set against fellow German Mona Barthel in the blink of an eye and then moving 4-1 ahead in the second. A late rally from Barthel, though, set up a tense finish before Beck eventually closed out the match, 61 64. In Monday's first main draw match, Peer won 64 64 against Petra Martic.