“We express our deep condolences to Mr Horn’s family and loved ones,” Philippine President Rodrigo Duterte’s spokesman Salvador Panelo said in a statement.
“We vow to pursue his killers to the ends of the earth until they are brought to justice.”
Abu Sayyaf has been blamed for some of the worst terror attacks in Philippine history, including repeated kidnappings of foreigners, who are often held for huge ransoms.
Philippine officials assert the group was behind the deadly January bombing of a Catholic cathedral on Jolo island during Sunday mass that was the worst attack to hit the nation in years.
The bombing was claimed by the Islamic State, which has worked to maintain a presence in the Philippines as its caliphate crumbled in the Middle East.
Abu Sayyaf was active in the Philippines years before linking up with Islamic State, and has supported its violent activities with kidnapping.
The group has held hostages over the course of years as it negotiated ransoms, but has also shown a willingness to kill its captives.
Abu Sayyaf beheaded German hostage Jurgen Kantner, 70, in 2017 after its demands for a roughly $600,000 ransom were not met.
Two Canadian hostages kidnapped from yachts moored at a marina on a tourist island in the southern Philippines were also beheaded in 2016 after demands for ransoms of similar amounts went unfulfilled.