Jonathan Calleri's first-half goal initially separated the two sides at the Estadio de Gran Canaria, but the away goal means that Valencia are favourites to make it through to the quarter-finals in the second leg on home soil.
The visitors thought that they had taken the lead inside the first 11 minutes when Ezequiel Garay sent the ball past the goalkeeper, but the referee blew up for a foul.
The tables turned when Las Palmas took the lead through Calleri, who connected with Borja Herrera's pass before sliding the ball into the bottom right corner.
Neither side peppered the respective goalkeepers with chances as the half went on, which was even more frustrating for the visitors given their status in La Liga.
The club currently sit third in the standings, while La Palmas are rooted to the bottom of the table, four points from safety after 17 games.
In the final 15 minutes, Valencia ramped up the pace, with Simone Zaza receiving a whipped-in ball into the box, which he met with a header but sent it over the crossbar.
Eventually, the equaliser came in the 85th minute when Rodrigo got on the end of Antonio Latorre's ball in the box before smashing it into the bottom right corner.
Zaza could have won the game for Valencia in the dying moments of the match had he not missed the target with a header, but his team remain in a good position to make it into the quarter-finals.
Las Palmas (4-2-3-1): Chichizola; Simon, Galvez, Bigas, Herrera (Navarro 73'); Aquilani (Castellano 63'), Samper (Toledo 88'); Tana, Viera, Momo; Calleri
Valencia (4-4-2): Domenech; Montoya, Garay, Vezo, Latorre; Pereira (Rodrigo 59'), Kondogbia (Parejo 46'), Maksimovic, Gil de Pereja Vincent; Mina (Guedes 66'), Zaza