British Airways says it has been able to operate extra flights as more cabin crew than expected crossed picket lines and turned up for work.
Union members launched a fresh five-day strike on Saturday, the third and last in their current series of walkouts.
The long-running dispute is over pay, staffing levels and working conditions.
With no sign of a breakthrough between BA bosses and the leaders of the Unite union, Unite has threatened to ballot for more strikes in July.
The airline has said it plans to fly more than three-quarters of customers who booked flights - which amounts to 65,000 people a day over the strike period.
It maintains that a growing number of strikers have been returning to work and that because of this, it was able to operate extra flights to Los Angeles, Washington, Mexico City and Phoenix from Heathrow on Saturday.
The airline has also said it expects to operate all flights to South Africa ahead of the World Cup.
These departures will be part of the 80% of long-haul flights it aims to operate from Heathrow next week, up from 70% during the previous walkout, which ended on Wednesday.
It also expects to run 60% of short-haul services from the airport, up from 55% during the previous five-day stoppage.
Flights from Gatwick and London City airports will continue to be unaffected.
Passenger numbers hit
Earlier this week, it emerged that the number of passengers British Airways carried in May fell by 14.2% from a year ago, as the strikes hit the airline.
BA carried a total of 2.37 million passengers in May, down from 2.76 million a year earlier.
Revenue passenger kilometres, a measure of how many full seats flew, were down 11.5%.
The financial impact of the strikes continued "at some £7m per day", BA said in a statement on Thursday.
"The total cost of the strike period can only be assessed at the end of the disruption and will reflect lost bookings and reduced travel volumes offset by some volume driven cost savings," it added.
Meanwhile, Unite has announced it will increase strike pay, with its members now receiving £45 a day, up from £30 in previous strikes. It is also considering offering interest-free loans of £1,000 for hardship cases.