BA said that flights over the next few days will still be disrupted because it is too late to unwind contingency plans already put in place to cope with the planned 20-day walkout by members of the Unite union.
But it said it aims to restore a full flying programme at London's Heathrow airport by the weekend, as long as the volcanic ash cloud that has shut airspace in recent days does not cause further problems.
The high court granted the last-minute injunction because of irregularities in Unite's ballot of workers for the action.
The strike had been due to begin at midnight and run in four blocks of five days each through early June.
BA has urged Unite - which can take the case to the court of appeal - to negotiate an end to the increasingly acrimonious dispute over pay and working conditions that had led to the planned walkout.
Before the court ruling, BA had said it planned to fly more than 70 per cent of its customers over the strike period, using leased aircraft and crew as well as BA cabin crew who decide not to take part in the walkout and staff reassigned from other jobs at the carrier.
The proposed strike dates - May 18-22, May 24-28, May 30-June 3 and June 5-9 - fell over a busy British school summer vacation period, a long weekend and the run-up to the football World Cup in South Africa, which begins on June 11.