Grammozis was named Schalke's fifth head coach this season on Wednesday (AEDT) with a contract that will see him in charge until June 2022, in theory at least.
However, the second season of that deal looks likely to be in the second division unless he can turn results around in a hurry.
This is his first coaching job in the Bundesliga.
The 42-year-old Greek-German now has 10 games left to bridge the eight-point gap needed to get Schalke off the bottom and out of the relegation places.
Schalke has just one league win all season and is on course to be relegated after 30 consecutive years in Germany's top flight.
The away point moved Mainz up one place to 16th and level on 18 points with Arminia Bielefeld and Hertha Berlin, which are also fighting relegation.
Mainz, which earned a shock win over Borussia Moenchengladbach a fortnight ago, peppered the Schalke goal with 19 shots on target, while the hosts managed just two against their opponents.
Mainz's best chance came eight minutes from time when 20-year-old striker Jonathan Burkardt forced a stunning reflex save from Schalke goalkeeper Frederik Roennow.
Former Arsenal defender Shkodran Mustafi came closest for Schalke when his powerful header ricocheted off the legs of Mainz goalkeeper Robin Zentner after only 18 minutes.
Schalke was dealt a blow before kick-off when playmaker Amine Harit dropped out with an injury.