The Canary Islands club, who are three points clear at the top of the Liga Adelante, beat top-flight Celta Vigo 2-1 at home in the first leg, giving themselves a good chance of securing a last-16 clash against Athletic Bilbao or third-tier Alcoyano when they play at Celta on Tuesday.
Like many Spanish clubs, Las Palmas have suffered severe financial problems in recent decades and came close to folding just under 10 years ago when they were playing in the third tier.
After emerging from insolvency proceedings and cleaning up their finances under a new president, they were promoted back to the second division in 2005-06 and this term are well placed to secure a berth in the top flight for the first time since 2002.
Paco Herrera, a former Celta coach who was an assistant to Rafa Benitez at Liverpool, has been in charge at Las Palmas since July and has won plaudits for an attacking style that has yielded 31 goals in 17 matches this term.
After Saturday's 4-2 win at Recreativo Huelva, Herrera played down the significance of the Cup game and urged his players, who include former Spain midfielder Juan Carlos Valeron, simply to enjoy the experience.
"It's the kind of game you just have to savour and I am not worried about the result," Herrera said at a news conference.
"But yes we do want to put on a good show," added the 61-year-old, who also had a stint in charge at Real Zaragoza.
Holders Real Madrid and Sevilla are already through to the last 16 and Real are almost certain to face city rivals Atletico, who have a 3-0 advantage over third-tier Hospitalet ahead of their second leg at the Calderon on Thursday.
The victors in a Real-Atletico clash are likely to meet record winners Barcelona in the quarter-finals.
Barca won their first leg at Huesca 4-0 and host the third-tier side for the return on Tuesday.
New Real Sociedad coach David Moyes will want to avoid an embarrassing upset when his side host third-tier Real Oviedo on Wednesday after a 0-0 draw in the first leg.
(Editing by Patrick Johnston)
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