The hacking of Kan's website did not affect the regular television relay of the show on Tuesday night in Israel or abroad.
Israel's hosting of the 2019 version of the European music competition has drawn pro-Palestinian boycott calls to protest against Israeli policies.
Hamas, a Palestinian Islamist group that controls the Gaza Strip, had no immediate comment.
Earlier this month it engaged in three days of fighting with Israel that included hundreds of rocket attacks from Gaza and Israeli air strikes in the enclave.
After the 41-country competition kicked off on Tuesday with a first semi-final, Kan's webcast cut to animated satellite footage showing explosions in Tel Aviv set to a menacing soundtrack.
Kan played down the hack, noting that the evening ended without further incident as Greece, Belarus, Serbia, Cyprus, Estonia, Czech Republic, Australia, Iceland, San Marino and Slovenia made it through to Saturday's finals.
"We know that at a certain stage there was an attempt, apparently by Hamas, to commandeer our digital broadcast," Kan CEO, Eldad Koblenz, told Israel's Army Radio.
"But I am happy to say that within a few minutes we managed to assume control over this phenomenon."
Palestinians and their foreign supporters have called, so far fruitlessly, on countries to shun the Tel Aviv songfest as part of wider efforts to isolate Israel internationally.
Many boycott calls have targeted Madonna, who arrived in Israel on Tuesday, ahead of the American pop star's guest performance at the Eurovision final.
The second semi-final is on Thursday.
Rebuffing the pressure, Madonna said she would "never stop playing music to suit someone's political agenda."
The European Broadcasting Union, which oversees the contest, did not have a number to hand for worldwide viewers of Tuesday's show. But an EBU spokesman said that, last year, a total of 186 million people watched both semi-finals and the finals.
Kan said on Wednesday that an Israeli truck driver injured a day earlier while unloading technical equipment in the parking lot of the Eurovision venue had died.
"Our thoughts and condolences go to his family," the broadcaster said in a statement, adding that it was working with authorities to ensure safety at the competition.