Passengers faced hours-long lines to check in, reclaim lost luggage or rebook flights at Terminal 5, BA's hub at Heathrow.
Cruz apologized in a video statement, saying: "I know this has been a awful time for customers".
BA cancelled all its flights out of Heathrow and Gatwick on Saturday after the IT failure, which shut down all of the carrier's check-in and operational systems and affected call centers and its website.
Data from flight tracker FlightAware.com showed BA canceled another 27 flights and had 135 more delayed Monday, a bank holiday in the United Kingdom that sees a high level of air travel.
Short-haul flights from the airport were subject to cancellations and delays, after the IT glitch led to scenes of chaos at Heathrow and Gatwick over the weekend.
The airline blamed the computer blackout on a "power supply issue" and said there was no evidence of it resulting from a cyber attack.
British Airways CEO Alex Cruz denied cost-cutting was the cause of a major IT outage that disrupted thousands of travellers and has said he will not resign over the incident. Many of the Irish airline's customers switch to long-haul services offered by BA from Heathrow.
"An IT failure is most likely the responsibility of the airline, and it is therefore legally required to compensate its passengers", says Eve Buechner.More news: SpaceX launches 1st recycled cargo ship
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The airline cancelled more short-haul flights from Heathrow yesterday but it has not explained the cause of the problem, which left large numbers of passengers sleeping in airport terminals.
He reassured passengers who had to return home without their bags, saying they would be sent by courrier.
Experts predict BA is facing a huge compensation bill, estimated at more than £100m.
She said that flight was cancelled, and another flight she rebooked for Sunday was also cancelled, while there was no sign or news of the couple's luggage.
"Work continues to restore all of our IT systems, but we expect some further disruption today", British Airways posted on its Facebook page Sunday morning.
Data from FlightAware showed two dozen British Airways and over a hundred Iberia flights were canceled Monday.
Three days of chaos caused by an IT glitch at Delta airlines past year was expected to cost the U.S. carrier $150m in lost revenue.