Luis Suarez's 44th goal in 64 matches for Barca looked to have earned them a seventh straight league win.
The home side, led by interim head coach Voro and Neville's brother Phil, were second best for long spells.
But the visitors paid for missing a host of chances as forward Mina, who turns 20 on Monday, lashed in late on.
It was the teenager's first goal since joining Valencia from Celta Vigo in the summer, and came after Paco Alcacer had chested down a long ball to set him up.
Plenty of spirit brings reward
Ruben Vezo was frequently left trailing by Neymar during a testing evening
Neville, who takes charge officially on Sunday, arrives at the Mestalla to take over a team languishing in the middle of La Liga after qualifying for the Champions League in May.
He will have been encouraged by a Valencia performance far better than was put forward in the 1-0 defeat at Sevilla last Sunday that preceded Nuno Espirito Santo's resignation.
This was a point earned against the odds. Valencia were missing 10 senior players for the visit of the Spanish champions, and it showed at times.
