The 1,000 Royal Marines and other personnel are being redeployed to central Helmand province.
The UK has suffered its heaviest losses in Sangin. Of the 337 UK deaths since 2001, a third have happened there.
Defence Secretary Dr Liam Fox said UK troops should be “proud of their achievements”.
He said Sangin was “one of the most challenging areas of Afghanistan”.
“The level of sacrifice has been high and we should never forget the many brave troops who have lost their lives in the pursuit of success in an international mission rooted firmly in our own national security in the UK,” he said.
British troops will redeploy to central Helmand, where they will continue to lead the fight against the insurgency and assist in building a stable and secure Afghanistan, he added.
UK forces have been in the area since 2006, and 106 UK personnel have been killed, 36 this year alone. The MoD announced in July that British troops were to be replaced by US forces.
Control was formally handed over from UK forces to the US Marine Corps at 0630 BST.
The BBC’s Ian Pannell in Kabul said there would be a physical handover, with the union jack lowered and the US flag raised, but little would change on a practical level.
He said some members of the 40 Commando Battle Group had already left, and the handover would be staggered over the coming weeks.
Describing it as a “totemic” moment for the UK, he said Sangin was the most dangerous district in Helmand – if not the whole of Afghanistan.
On a recent visit to the area, he witnessed a long battle in which a number of US soldiers, Afghan soldiers and civilians sustained injuries.
“Although progress has been made, the area remains very difficult. It is a key battleground for insurgents and coalition forces.
“The truth is, the Americans will now have to try and finish the job that Britain started,” he added.
Ian Sadler, whose son Jack was killed when a landmine exploded underneath his army Land Rover near Sangin in 2007, said he was glad troops were getting out of the area.
“We are pulling out of a hotspot. It’s a particularly dangerous area of Afghanistan, so it’s a good thing,” he told the BBC.
“I think it is a shame that while our soldiers were in Sangin they did not have the best vehicles that could have been provided and I still don’t think there’s enough helicopters.”
The former serviceman said it was part of a soldier’s job to keep moving on.
“They will be sad for the ones who have been killed, for those who have lost their arms and legs,” he added. “But Sangin is like staying at a bad hotel – you’re glad when you don’t go back.”
Col Stuart Tootal, former commander of 3 Para, the first battle group sent into Sangin, says the area will always be significant for British forces.
“We can’t ignore the emotion the British are going to attach to Sangin. I mean my own battle group went in there four years ago and half our casualties were lost in Sangin,” he said.
The handover “makes pragmatic good sense because it allows the British to focus on their main effort in the centre of the province,” he added.
MoD spokesman Maj Gen Gordon Messenger, a former commander of the UK Helmand task force, said it was “absolutely not” a pull-out .
While progress had been slower in Sangin than in other parts of Helmand, he admitted, British efforts had not been worthless.
“It’s a hugely important point that the Afghan flag flies in the district centre of Sangin, and that the people of Sangin and the surrounding area recognise that and see the benefits of it,” he said.
“By contesting Sangin, by showing improvement in Sangin, we’re able to deflect violence that would otherwise by exported into the populated areas in central Helmand.”
The commanding officer of 40 Commando group, Lt Col Paul James, said the handover was a “poignant moment” tinged with sadness, but the overwhelming emotion was one of pride.
“I think we’ve achieved significant success here, making Sangin a much more stable and peaceful place.
“And probably just as importantly, the Afghan national security forces that we’ve partnered here are now starting to be able to stand on their own two feet and take on the responsibility for delivering Sangin for themselves.
“It’s not going to be British forces who deliver success in Sangin and it’s not going to be American forces.
“It’s not going to be anyone else other than the Afghans themselves.”
Lt Col Clay Tipton, from the US Marine Corps, said UK and US troops had been working together in Sangin for the past couple of months, and their job now was to “continue the success”.
The BBC’s Quentin Sommerville in Kabul says Sangin is a town of about 20,000 people and the area is a vital supply route for the Taliban, with men, money and drugs passing through it.
He said British patrol bases were already being shut down and the Royal Marines who served there are frustrated at having to leave a place they fought so hard for.
“As the British mission in Sangin ends, questions remain whether there were ever enough resources to complete the job,” he said. “Even so, the American force will cover the same amount of ground, with a similar number of troops.”
There are about 9,500 UK troops in Afghanistan, with the majority deployed in the south of the country. -BBC