Substitute Macauley Chrisantus struck an 84th-minute equaliser to ensure Las Palmas would head to Seville for next month's second leg on level terms.
Ruben Castro had fired Betis, fifth in La Liga, ahead on 67 minutes.
Real Zaragoza grabbed the advantage in their all-La Liga tie against Levante with a 1-0 win in Valencia.
Carlos Aranda waited until injury time to snatch the only goal of the game.
Thursday's Copa del Rey Results
Real Betis 1-1 Las Palmas
Levante 0-1 Real Zaragoza