Afghanistan is currently counting the votes of its presidential and provincial elections. The security situation has been deteriorating and the death toll of foreign troops is at a record high.
The UN says the drugs trade, which helps fund the insurgency, threatens the legitimacy of the Afghan state.
Chris Morris, BBC News, Kabul
One of the things the UN is extremely concerned about is what it calls the ticking bomb of huge stockpiles of opium in Afghanistan - enough, it says, to fuel two year's supply of world heroin.
The fact that there are these huge reserves out there somewhere, coupled with prices lower than they have been for 10 years, means that market forces are clearly pushing opium cultivation down. And that could always be reversed. The hope is that this is the start of a long-term downward trend, but it cannot be guaranteed.
The UN is still worried about the corrosive effect of corruption linked to the drugs trade, and it says more needs to be done to persuade farmers that taking on alternative livelihoods can be to their benefit. So this is a window of opportunity, but it is nowhere near the end of the fight.
The UNODC report called on the international community to sustain progress in Afghanistan, which produces 90% of the world's heroin.
It praised the introduction of UK-backed "food zones", which distribute wheat seed for farmers to plant.
The report said 20 of the country's provinces were now poppy-free, but the BBC's Chris Morris in Kabul says that does not mean they are free of the refining and trafficking of drugs.
The UN report said that the price of opium was at a 10-year low.
The UNODC's executive director, Antonio Costa, said: "The bottom is starting to fall out of the Afghan opium market. These results are a welcome piece of good news and demonstrate that progress is possible."
But he warned that Afghan drugs still have catastrophic consequences: funding criminals, insurgents and terrorists, encouraging corruption and undermining public trust.
Some analysts suggest that - as world heroin prices remain low - the fall in opium production may simply be a temporary tactic by suppliers to drive prices back up.
Thomas Schweich, who was US ambassador for counter-narcotics in Afghanistan from 2007-08, conceded there might be "a little bit of truth" to that.
But in an interview with the BBC he insisted that nonetheless, the overall trend remained a downward one that represented a victory for eradication strategies.
"It's a sustainable trend, not a one-year anomaly," said Mr Schweich.
This year there were 69,833 hectares devoted to poppy growing in Helmand, a sharp fall from 103,590 hectares in 2008, the report found.
However, this year's figure was also more than double the 26,500 hectares used for poppy growing in the province in 2005, the year before British troops deployed in the area.
Helmand continues to account for nearly 60% of the country's total production of the drug, the UNODC report said.
Mohammed Jawad, a farmer in the Helmand city of Lashkar Gah, told the BBC he would grow poppy if he could get away with it.
"This year I'm growing wheat, tomatoes, potatoes and watermelon," he said. "The government is not letting us grow poppy. If I was further away, I would grow poppy. I know growing poppy takes a lot of work, but what can we do?"
Ahead of Wednesday's meeting in Paris by European and American officials to discuss Afghanistan, US envoy Richard Holbrooke told French media that a summer offensive in the south to flush out the Taliban had yielded results.
"The coalition forces including the British and Americans have done vast damage to the Taliban, disrupted them, captured major caches of opium, heroin and drug paraphernalia," Mr Holbrooke told France 24 television.
Foreign troops fighting in Afghanistan endured their deadliest month yet in August when 77 soldiers died, according to an independent website - icasualties.org - which tracks military casualties.
More than 300 troops have died in Afghanistan so far this year, making it the deadliest year since the 2001 US-led invasion to overthrow the Taliban.
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