The tie was delicately poised entering the second leg following a 1-1 draw in the reverse at the Mestalla last week. Gary Neville's side stepped aside from their league troubles, however, to advance courtesy of a first-half effort from Rodrigo.
Neither side created a chance of note inside the opening 10 minutes, but it was Valencia that looked most likely to make something happen in the final third of the field.
Down the other end, Willian Jose's pace saw the Las Palmas striker face up to Mathew Ryan inside the Valencia box, but the Los Che goalkeeper did enough to force the Brazilian away from the goal, before Ryan kept out a powerful free kick from the same player moments later.
It was Valencia that took the lead on the night and indeed in the tie, however, with Rodrigo converting a low cross from Antonio Barragan in the centre of the Las Palmas box.
Momo came close to levelling the scores in the 25th minute when he showed excellent feet outside the Valencia box, but the No.11's final strike was just deflected wide of the post. Momo then turned provider for Las Palmas skipper David Garcia 10 minutes before the break, but the central defender headed over from inside the penalty area.
Valencia came close to scoring their second in the final moments of the first period, but Dani Parejo's rasping free kick was just pushed over the crossbar by Raul Lizoain, before Shkodran Mustafi headed the resulting corner wide of the goal.
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Willian Jose tested Ryan with a strike from distance early in the second period, but the Australian international once again got down to make the stop, before keeping out Momo from a similar distance.
Negredo remained Valencia's biggest threat, meanwhile, and the Spaniard forced Lizoain into a smart stop after meeting a cross from Gaya. Rodrigo then had a golden chance to make it 2-0 to the visitors just before the hour - the attacker lifted his effort over Lizoain, but also over the crossbar.
Down the other end, Momo had an even better chance, but the attacker found Ryan in sensational form once again.
Las Palmas were the team firmly in the ascendancy as the match approached its latter stages and Valencia did not help themselves with some sloppy work in possession of the ball. Aythami Artiles had an outstanding opportunity to level for the home side in the first minute of stoppage time, but the centre-back somehow headed wide of the post from close range.
Los Che continued to face pressure as Las Palmas looked for the goal that would have forced extra-time, but Neville's side managed to hold out - booking their spot in the final four of the Spanish Cup alongside the holders Barcelona, Celta Vigo and Sevilla.