The specialist insurance market, made up of 88 underwriting syndicates, said the bill was "well within" the worst-case scenarios it prepares for.
Chief executive Richard Ward said: "Our priority is to pay valid claims as quickly as possible and help communities in North America and the Caribbean affected by Sandy get back on their feet."
Lloyd's said its estimate of between $2bn and $2.5bn was consistent with wider industry losses of between $20bn and $25bn.
Sandy, which ravaged America's north-east coast, is expected to be the country's second-costliest storm after Hurricane Katrina in 2005. That storm left Lloyd's facing claims worth $4.3bn.
The market has shown in recent years it is more than able to cope with major catastrophes and met its own claims in 2011 without any call on its central fund – its fund of last resort.
It saw a profit of £1.5bn in the first six months of this year, compared with a loss of £697 million a year earlier, the second-most- expensive year on record for the insurance industry.
The market saw £12.9bn claims in 2011, including £4.6bn related to disasters such as floods in Australia and Thailand, an earthquake in New Zealand and the tsunami in Japan.
Mr Ward added: "The market's total exposure is well within the worst-case scenarios we prepare for."