Neville left the club this week after less than four months in charge and was replaced by Pako Ayestaran - a man Neville had brought to the Mestalla.
Valencia went ahead when Rodrigo scored after goalkeeper Javi Varas's error.
But Jonathan Viera's penalty and an own goal from Valencia's Shkodran Mustafi gave Las Palmas the three points.
The result leaves Los Che 15th in La Liga, only six points above the relegation zone with seven league matches left this season.
Valencia, who finished fourth in the table in 2014-15, have not played in the second tier of Spanish football since 1986-87.
Valencia lost in their first game since the sacking of former boss Gary Neville as they suffered an away defeat at Las Palmas to remain in relegation trouble.