But a hard core of the 3,000 names present said they would become "refuseniks" and fight on in the courts.
David Rowland, chairman of Lloyd's, began the annual meeting at the Royal Festival Hall in London secure in the knowledge that proxy votes had already put him well on the way to success.
The votes were only the first stage in a series culminating in a poll of all members that closes on 28 August. But without the approval of members for a pounds 440m special contribution to the rescue fund, the plan would have fallen apart.
Rebel names claimed they had enough continuing support to finance new legal actions for fraud against the market, though Lloyd's officials were dismissive of the numbers claimed to be ready to follow them.
The exact total will not be known until the August deadline for members to accept or refuse their individual offers, though the votes yesterday were regarded as indicating almost certain success for the rescue plan.
Alan Porter, a Lloyd's member, claimed there would be between 4,000 and 5,000 "refuseniks". "The case for fraud in the form of bad faith and reckless misrepresentation has gathered sufficient strength to be endorsed for the first time by leading counsel in the last few days." Lloyd's should be pursued until the full truth came out, he added.
Mr Porter, chairman of the property company London and Industrial, was the principal speaker on behalf of four motions from rebel names which attempted to force Lloyd's to improve its offer by as much as pounds 1bn. The rebel names had requisitioned an extraordinary meeting to follow the annual meeting.
Ron Sandler, chief executive of Lloyd's, said "any attempt to implement these proposals must lead quickly to the collapse of Lloyd's and I can't help but feel that some of the sponsors of these resolutions understand that only too well".
Even before the votes on the rebel motions were taken, Mr Rowland revealed that 85 per cent of the proxies he had received were against.
After the votes on the rebel motions, Lloyd's broke up into three separate meetings to approve the crucial pounds 440m payment towards the rescue.
It emerged that by the start of these later meetings between 94 and 98 per cent of the proxy votes submitted were in favour.