TUIfly was also forced to cancel flights past year after many pilots called in sick, with their union saying staff were concerned that merger talks, since failed, could lead to job and pay cuts.
Both domestic and global flights were affected, including links between Berlin and Los Angeles or Dubai.
"We worked hard to be able to make this offer, with help not only from experts but also supportive comments and suggestions from employees, passengers and business partners of Air Berlin", the Bavarian airline tycoon wrote in a Facebook post late Sunday.
Mr Winkelmann said: "This day is costing us several million euros".
The company had to cancel more than 110 flights out of a planned 750 on Tuesday because of the walkout. It is essential that operations be stable in order for these negotiations to go well.
Air Berlin filed for insolvency in mid-August, after its main shareholder Gulf carrier Etihad pulled the plug on its cash lifeline.
Air Berlin continues to operate due to a 150 million euro ($180 milliom) credit loan from the German government.
"That's the only way we can secure as many jobs as possible", he added.
Pilot union Vereinigung Cockpit says it is "surprised" by the cancellations and stresses that it has not called on its members to call in sick.
They announced the cancellation of more than 100 flights on Tuesday.
Other interested airlines cited in media reports include package holiday firm TUI, British low-priced carrier EasyJet, Thomas Cook subsidiary Condor, as well as Bavarian entrepreneur Hans Rudolf Woehrl.