A number of Maersk's 76 container terminals were affected and were forced to run on manual systems, AP Moller Maersk chief operating officer Vincent Clerc told AFP, refusing to specify which terminals were impacted because of the "fluidity of the situation."
"Some terminals that were down this morning are now up and running," Clerc said.
Maersk's two terminals in Rotterdam, Europe's biggest port, were however "still affected" on Wednesday, Clerc said.
The port, one of the top 10 in the world, handles more than 461 million tonnes of cargo a year, and welcomes the largest container ships in the world.
APM Terminals, part of the AP Moller Maersk conglomerate, runs the two terminals at the sprawling port which stretches across 42 kilometres.
India's shipping ministry said meanwhile a terminal run by Maersk at the Mumbai port, the largest in India, was also affected.
While the systems are down, "we have to manage on a manual basis... It's difficult for people in the terminals to tell the people on the ground – the longshoremen -- which containers to unload," Clerc said.
A spokesman for APM Terminals in Rotterdam, Tom Boyd, said the manual process was tough work.
"Today we are handling 4,500 containers. It's more labour extensive, but we are making it work. We are communicating with our customers through gmail and other things because the IT system is down," he told AFP.
Maersk said meanwhile that all new orders were also on hold.
"Today we have not taken any new orders to our platforms. Customers are booking through third parties but not through our platforms as a precaution," Clerc said.
"We are working on being able to open up for new orders from tomorrow, probably through alternate processes," he added.
The series of cyberattacks began in Russia and Ukraine on Tuesday, hitting government and corporate computer systems across the world as the virus spread to western Europe and across the Atlantic.
Several other multinational companies said they were targeted, including US pharmaceutical giant Merck, Russian state oil giant Rosneft, British advertising giant WPP and the French industrial group Saint-Gobain.
Clerc said the cyberattack was "still ongoing" at Maersk on Wednesday at 1300 GMT, and that there had been a ransom demanded for the release of data but the Danish group had refused to pay.
"We have it contained ... by proactively shutting down systems and computers to prevent it from further contaminating our systems."
He said Maersk had not yet estimated the losses caused by the attack.
"A lot of it will depend on how quickly we can put in place our recovery plan... The longer we are affected the more the price tag will increase."