A first-half own-goal from Wilfried Zahibo sent the visitors ahead, but Paco Alcacer restored parity in the 61st minute to leave the tie delicately poised ahead of the second leg in Gran Canaria next week.
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Valencia had a wonderful chance to open the scoring in the sixth minute when Rodrigo robbed possession outside the Las Palmas box before sliding a pass in to Alcacer, but the striker's effort was tipped over by Raul Lizoain.
Aythami Artiles then put his goalkeeper in all sorts of bother with a short back-pass minutes later, but Lizoain was quick off his line to hack clear under pressure from the lively Alcacer, who captained Los Che on the night.
Lacklustre defending from Las Palmas handed Valencia a number of opportunities to create in the opening 30 minutes, but the final pass from Neville's side was too often wide of the mark.
A header from Rodrigo almost caught out the visitors as the half entered its latter stages, but Liazon was equal to the danger and on hand to make a smart one-handed save.
Instead, it was Las Palmas that took the lead in the 38th minute courtesy of an own-goal from 22-year-old midfielder Zahibo, who spectacularly headed past his own goalkeeper from a deep free kick.
There were more whistles than there were cheers when Valencia returned to the field for the second period, but Neville resisted the temptation to make changes, even with Alvaro Negredo on the bench.
Alcacer had a half-chance to level in the 48th minute after a smart cross from Pablo Piatti, but the Spaniard could only head wide of the post as the visitors remained ahead. Lizoain almost gifted Valencia an equaliser moments later when he failed to make contact on a back-pass, but luckily for the stopper, the ball just trickled wide of the post.
Alcacer remained Valencia's biggest threat and again came close in the 53rd minute when he met a cross from Piatti, but his effort flew over the crossbar, before Emmanuel Culio tested Mathew Ryan down the other end.
Valencia were level just past the hour, however, and it came from their talisman Alcacer, who found a yard outside the box before sweeping an effort into the bottom corner.
Jose Gaya had an opportunity to make it 2-1 Valencia in the 65th minute after a set-up from Rodrigo de Paul, but the left-back, who returned from injury for this match, could only fire over the crossbar.
The home side pressed for a winner late on, with Joao Cancelo's pace in a wide position creating a number of half-chances, while Gaya also set up Negredo from the opposite side, but Las Palmas held firm to secure a draw ahead of the return.