Eighth seed Delbonis lost serve twice in the first set against Roberto Carballes Baena but recovered well to claim a 7-5 6-1 victory in Marbella.
He will take on Norbert Gombos, who recovered to beat Juan Ignacio Londero 6-7 (3-7) 6-1 6-3, for a place in the quarter-finals of the clay-court event.
Kwon was made to do things the hard way against junior tennis world number one Rune, who took the match all the way.
Despite trailing 3-1 in the third set, seventh seed Kwon held his nerve to force a tie-break and prevailed 6-3 3-6 7-6 (7-4).
Ricardas Berankis is also through thanks to a 6-4 6-7 (5-7) 7-5 victory over Lucas Pouille in just under three hours.
Meanwhile, seeded pair Tommy Paul and Guido Pella fell at the first hurdle in the Sardegna Open to Yannick Hanfmann and Egor Gerasimov respectively.
Eighth seed Paul recovered after squandering a 4-1 lead in a first-set tie-break but fell 7-6 (10-8) 5-7 6-3, while seventh seed Pella lost 6-4 6-0 in straight sets to Gerasimov.
Elsewhere in Cagliari, Aljaz Bedene rallied past teenager Giulio Zeppieri 1-6 7-5 7-6 (7-3) and Laslo Djere eliminated Italian wildcard Federico Gaio 6-4 6-2.
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