Wozniacki, the seventh seed, lost inside two hours as she was edged out in a topsy-turvy first set despite assuming a 3-0 early lead, eventually losing it 8-6 in an extended tie-break.
But the world number 10 folded alarmingly in the second as she mustered just a single game with Suarez Navarro able to coast through to the second round.
It was a fine performance from the world number 23, who capitalised on a shaky display on serve from the Dane – who committed three double faults - and found 76 per cent of first deliveries herself.
Suarez-Navarro wrapped up her sixth career win over a top 10 player on her first match point when Wozniacki's baseline shot was ruled out.
"I played well and am very happy," Suarez Navarro said. "Caroline is a very good player and fights. It was a difficult first set, very close."
Next up for Suarez Navarro, who has reached four WTA finals but has yet to lift a trophy, will be a second-round encounter with Yaroslava Shvedova.
And sixth seed Samantha Stosur was also dumped out of the event in the final match of the day, losing 6-4 6-4 to Jelena Jankovic.
The Serb saved the only break-point chance her Australian opponent had while securing a break in each set to earn a spot in the second round.
Fifth-seeded Petra Kvitova of the Czech Republic did progress but only after struggling to beat wild-card entry Annika Beck of Germany 7-5 6-7 (0) 6-3.
WTA Stuttgart results
5-Petra Kvitova (Czech Republic) beat Annika Beck (Germany) 7-5 6-7(0) 6-3
Jelena Jankovic (Serbia) beat 6-Samantha Stosur (Australia) 6-4 6-4
Carla Suarez Navarro (Spain) beat 7-Caroline Wozniacki (Denmark) 7-6(6) 6-1
Yaroslava Shvedova (Kazakhstan) beat Roberta Vinci (Italy) 6-4 6-2
Lucie Safarova (Czech Republic) beat Mona Barthel (Germany) 6-4 6-1
Anastasia Pavlyuchenkova (Russia) beat Ekaterina Makarova (Russia) 6-7(5) 7-5 6-4
Bethanie Mattek-Sands (U.S.) beat Yanina Wickmayer (Belgium) 6-2 6-1
Julia Goerges (Germany) beat Kirsten Flipkens (Belgium) 6-2 2-1 (Flipkens retired)
Marrakech Grand Prix results
3-Alize Cornet (France) beat Karin Knapp (Italy) 1-6 7-5 6-0
Chanelle Scheepers (South Africa) beat Maria Teresa Torro (Spain) 6-2 6-1
6-Francesca Schiavone (Italy) beat Petra Martic (Croatia) 7-6(8) 6-4
Olga Puchkova (Russia) beat Daniela Hantuchova (Slovakia) 3-6 7-6(5) 7-5
Yulia Putintseva (Kazakhstan) beat Mathilde Johansson (France) 6-3 6-4
Flavia Pennetta (Italy) beat Melinda Czink (Hungary) 6-1 7-6(2)