The airline said it was running a full schedule at London's Gatwick airport on Monday and planned to operate all its long-haul flights from Heathrow, although some short-haul flights had been canceled.
BA had been forced to cancel all its flights from Heathrow, Europe's busiest airport, and Gatwick on Saturday after a power supply problem disrupted its operations worldwide and also hit its call centers and website.
The disruption continued on Sunday. Some stranded passengers curled up under blankets on the floor or slumped on luggage trolleys, images that played prominently in the media at the start of a week when schools were on holiday.
"Apologizes all well and good but not enough. BA has lost another loyal customer #disgraceful," tweeted Tom Callway, who had been due to fly to Budapest.
Spanish-listed shares of parent company IAG, which also owns carriers Iberia, Aer Lingus and Vueling, dropped 2.7 percent on Monday after the outage.
Davy analyst Stephen Furlong said the cost to the carrier of cancelling one day of operations was around 30 million pounds ($38.5 million) in revenue and 4 million in operating profit.
On top of that, the airline will pay compensation to customers for the delays, though he added it looked likely to be a one-off cost which would be limited given the resumption of flights on Sunday and Monday.
BA has been cutting costs to respond to competition on short-haul routes from Ryanair and easyJet and recently faced criticism for starting to charge passengers for their in-flight snacks.
Ireland's Ryanair was quick to seize on the marketing opportunity, tweeting "Should have flown Ryanair" with a picture of the 'Computer says no' sketch from the TV series "Little Britain" to poke fun at BA.
The GMB union said that BA's IT systems had shortcomings after they made a number of staff redundant and shifted their work to India in 2016.
"This could have all been avoided. BA in 2016 made hundreds of dedicated and loyal IT staff redundant and outsourced the work to India," Mick Rix, GMB National Officer for Aviation, said.
In response, BA said its IT services were provided by a range of suppliers and that this was common practice.
"We would never compromise the integrity and security of our IT systems," a BA spokeswoman said.
Several passengers complained about a lack of information from BA staff at the airport. Others said their luggage had been lost.
The airline said it was working to get reunite passengers with their luggage after many items were left at Heathrow over the weekend, although staff on Twitter warned this "could take some time".