"Our IT systems are now back up and running and we will be operating a full flight schedule at Heathrow and Gatwick," the airline said.
But it warned it "may take some time" to reunite travellers with their bags.
BA chief executive Alex Cruz earlier blamed a power surge for the disruption which affected 75,000 people.
"There was a power surge and there was a back-up system, which did not work at that particular point in time.
"It was restored after a few hours in terms of some hardware changes... we will make sure that it doesn't happen again," Mr Cruz said in his first interview since the systems failure.
He told the BBC this had affected "all the operating of our systems - baggage, operations, power processing".
The GMB union blamed the problem on technical staff being outsourced from the UK to India, a claim denied by Mr Cruz.
The focus is now likely to shift towards the financial impact on the airline.
BA is liable to reimburse thousands of passengers for refreshments and hotel expenses, and travel industry commentators have suggested the cost to the company - part of Europe's largest airline group IAG - could run into tens of millions of pounds.
Shares in  IAG, which is listed in both London and Madrid, fell 2.8% in Spain on Monday and are expected to fall again in London trading when it reopens after the bank holiday.
Davy analyst Stephen Furlong said the cost to the carrier of cancelling one day of operations was around Â£30m.
Questions remain about how a power problem could have had such impact.
The BBC's technology correspondent Rory Cellan-Jones said one theory was that returning systems were unusable as the data had become unsynchronised.
British Airways is now operating a full flight schedule after an IT failure saw the airline cancel thousands of flights over the bank holiday weekend.