Neville is still waiting for a first league win since taking charge of Valencia in December.
Former Barcelona B striker Sanabria, on loan from Roma, scored his fourth goal in two games after Danilo tripped Jony.
Alvaro Negredo missed three chances to equalise, heading one against the post, then chipping wide and blazing over.
Valencia, previously unbeaten at home in the league, then missed another great chance as Rodrigo Moreno and Sofiane Feghouli narrowly failed to connect with Zakaria Bakkali's left-wing cross.
At the other end, Australian goalkeeper Matty Ryan was fortunate not to concede a second penalty after barging into Carlos Carmona.
Valencia have not won any of the eight La Liga matches since former Manchester United and England defender Neville took charge and have gone a total of 11 league matches without a win, two short of equalling a club record.
Neville must attempt to lift his side in time for Wednesday's Copa del Rey semi-final first leg against Barcelona at the Nou Camp.
Gary Neville's Valencia were booed by their own supporters after Antonio Sanabria's penalty eased Sporting Gijon's relegation worries.