The airline’s site was down for at least seven hours, replaced by a message from the hacker group, before the company brought it back online by mid-afternoon in Malaysia.
The hackers at first changed the site to display a message saying "404 - Plane Not Found" and that it was "Hacked by Cyber Caliphate," with a photo of one of the airline's Airbus A380 superjumbo jets. The browser tab for the website said "ISIS will prevail."
Malaysia Airlines is struggling to recover from twin disasters last year, including the disappearance of Flight 370, which authorities believed crashed 1,800 kilometers (1,100 miles) off Australia's west coast, and the downing of Flight 17 over Ukraine.
The hackers later replaced the jet with a picture of a lizard in a top hat, monocle and tuxedo smoking a pipe.
The IS reference was removed and the claim of responsibility changed to "Lizard Squad - Official Cyber Caliphate," with a link to the group's Twitter account.
The airline said in a statement that it was a "temporary glitch" that didn't affect passenger bookings and that the breach had been reported to Malaysia's transport ministry and Internet security agency. It said user data "remains secured."
The group, however, tweeted that it was "going to dump some loot found on malaysiaairlines.com servers soon," and posted a link to a screenshot of what appeared to be a passenger flight booking from the airline's internal email system.
Going to dump some loot found on http://t.co/D9XYneQoaK servers soon— Lizard Squad (@LizardMafia) January 26, 2015
The particular booking was made by Malaysian Amy Keh, who said she had made it in October for her mother and two relatives to travel from Kuala Lumpur to Taiwan in March.
"I am a bit worried about their security. Now the whole world knows that they will be going to Taipei," said Keh, who logged in on Monday to check the itinerary. She said the website looked different and called the airline, which told her of the hacking. However, she only found out when contacted by The Associated Press that the travel information was posted online
The Lizard Squad group last year claimed it was behind attacks on Sony's online PlayStation network and Microsoft's Xbox site.
In August, it also tweeted to American Airlines that there might be explosives on a plane carrying the president of Sony Online Entertainment, which makes video games, forcing the flight to be diverted.
Explaining how the hack had occurred, Malaysia Airlines said its domain name system was "compromised" and users were redirected to the hacker group's website. The domain name system translates web addresses typed into browsers into the numbers that computers use to identify and connect with each other on the Internet.
The IS group now holds about a third of both Syria and Iraq, territory it has declared a caliphate. Police in Malaysia detained 43 people last year on suspicion of links to the extremist group, underscoring concerns held by Prime Minister Najib Razak that the spread of Islamic State ideology could lead to conflict in predominantly Muslim Malaysia.