Puri's comments came after a UN Security Council meeting in New York to discuss the crisis.
India and Pakistan have have been trying to mend ties after the Mumbai attacks in 2008 by a Pakistan-based armed group put relations between the two neighbours at a new low.
'Years of need'
The latest UN figures showed that $490.7m has been raised for relief so far, with another $325m pledged.
This total surpassed the $460m the UN asked for last week.
The US has donated the most, followed by Saudi Arabia and Britain.
Ban Ki Moon, the UN Secretary General, welcomed the donations, saying: "The generosity of countries and individuals will make a real difference in the daily lives of millions of people.”
"We must keep it up. Pakistan is facing weeks, months and years of need."
Maurizio Giuliano, the spokesman for the UN Office for the Co-ordination of Humanitarian Affairs (OCHA) in Islamabad, said that the need for donations may sharply increase since the UN's last estimate on August 11.
"The number of people in need of immediate humanitarian aid has risen from six to eight million," Giuliano told AFP news agency.
"We have more than doubled the rate at which we are delivering relief but, since August 11, the number of people who need emergency help has undoubtedly more than tripled. We are in a race against time."
The UN will have to revise its target within 30 days following the launch of the appeal, he added.
There are already over 38,000 cases of acute diarrhoea and at least one case of cholera has been confirmed. A major disease breakout would cause another crisis and impose new demands on already stretched humanitarian workers.
The official death toll is around 1,500 but the true number of people killed in the disaster may turn out to be higher, with large areas of the country still inaccessible.
The floods began on July 29 in the northwest of the country after exceptionally heavy monsoon rains and have since swamped thousands of towns and villages in Punjab and Sindh provinces.
While rainfall has lessened, flooding is continuing in parts of Sindh province as water from the north courses down the Indus and other rivers.
The Nato military alliance had agreed to provide an airlift and sealift for aid to Pakistan and said the first cargo plane would fly there at the weekend with power generators, water pumps and tents.
Many areas are inaccessible by road and the only way to provide relief to them is through helicopters.