Sharapova let a big second-set lead slip away, then needed four match points to close out Schnyder 6-2, 7-6 (6).
Serena Williams is ranked 183 after her pregnancy absence and French Open chiefs decided against seeding her.
Now ranked 24 in the world, Sharapova heads to Wimbledon without any competitive grasscourt action.
Former world number one Williams, a three-time champion in Paris, is playing a Grand Slam for the first time since winning the Australian Open in 2017.
Main-draw action begins Sunday in what would be Williams' first Grand Slam tournament in more than a year.
Svitolina, the fourth seed, received a first round bye and dropped just three games in a dominant 6-1, 6-2 victory over 35th-ranked Petra Martic.
Sharapova giggled with delight as she revealed how she plucked up the courage to pop the question and got to practise with Nadal.
Simona Halep retains top spot in WTA rankings, Petra Kvitova moves to eighth following Madrid Open win
Petra Kvitova jumped two spots to eighth in the latest WTA rankings released after beating Kiki Bertens to win the Madrid Open.
After serving a 15-month drugs ban for use of the illegal heart medication meldonium, Sharapova's return to the WTA circuit began in Stuttgart exactly a year ago.
Sharapova has decided to split with her coach Groeneveld following her shock first round exit from the Indian Wells.
The former world number one Sharapova was one of the tournament's major attractions after missing the last two events.
All you need to know about Meldonium, the drug at centre of Maria Sharapova, Russian curler Alexander Krushelnitsky controversies
Also known as mildronate or mildronats, meldonium can increase endurance and recovery. It was only added to the World Anti-Doping Agency's (WADA) list of banned substances at the start of 2016.
Sharapova, one of the highest earners in women's sport, suffered a string of early defeats after her comeback last April.
The 2016 winner from Germany and 21st seed roared past the unseeded Russian 2008 champion 6-1, 6-3.
An FIR has been lodged against tennis icon Maria Sharapova for endorsing a housing project in Gurugram which never took off.