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 program at London's Heathrow airport by the weekend, as long as the volcanic ash cloud that has shut airspace in recent days doesn't 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.
It is the second time in less than six months that BA has successfully averted a strike by Unite by claiming voting irregularities in a High Court claim; a planned Christmas and New Year walkout was halted the same way.
BA 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.
"We hope all sections of Unite, including the leaders of the cabin crew branch Bassa, will take this opportunity to pause and focus on achieving the early and peaceful end to this dispute which the travelling public and all our employees want," the airline said in a statement.
Unite, which represents around 90 percent of BA's 12,000 cabin crew, had no immediate comment.
The dispute between BA and its workers shows no sign of a breakthrough, with no positive reports from separate talks on Monday between union leaders, airline management and government officials.
The union pushed ahead with a strike in March despite a new offer from the airline, costing BA around 45 million pounds ($65 million). BA retaliated by taking away staff travel perks and taking disciplinary action against some 50 workers, leading Unite to accuse airline management of intimidation.
Walsh said earlier there was "absolutely no way" he would intervene in disciplinary cases that the union said were now holding up the prospect of a deal.
BA said that no staff had been suspended for going on strike. Of 27 individuals that had been investigated after allegations mainly of bullying and intimidation, seven were dismissed for series cases of misconduct, the airline said. Another 20 had returned to work, 15 of whom received written warnings, it added.
Before the court ruling, BA had said it planned to fly more than 70 percent of its customers over the strike period, using leased planes 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.
Walsh has repeatedly warned that the disputed changes to work practices, including fewer staff on long-haul flights and a yearlong pay freeze, are necessary for BA to survive in the post-financial crisis climate.
The company has been hit hard by the downturn because of its heavy reliance on premium fare passengers on the trans-Atlantic route. As many leisure and business travelers seek lower cost options, there are fears that its core business will never fully recover.
Along with other airlines, it also booked big losses last month when an Icelandic volcano eruption caught the industry unawares, closing down European airspace for days as an ash cloud drifted across the region.