Lancashire's loss to Warwickshire meant Hampshire needed to win to survive.
They declared on 245-9 to leave Durham 296 from a minimum of 78 overs for victory, which they reached to end the hosts' two-year stay in the top tier.
Mark Stoneman (137) and Scott Borthwick (88) excelled in their last innings for Durham before they join Surrey in 2017.
England all-rounder Ben Stokes (23 not out) struck the winning runs for the visitors.
Hampshire had opted to bat on during the morning session to extend their lead close to 300 and, with Lancashire struggling at Edgbaston, it soon became clear that only a win would be enough for survival.
Ryan Pringle's 7-107 indicated that Hampshire's spinners would be their best hope of success, but a second-wicket stand of 162 between Stoneman and Borthwick ended their chances of achieving safety.

The reactor at Yongbyon has been the source of plutonium for North Korea's nuclear weapons programme.
The White House said North Korea should "focus instead on fulfilling its international obligations".
The reactor was shut down in 2007 as part of a disarmament-for-aid deal.
But Pyongyang vowed to restart it in 2013, following its third nuclear test and amid high regional tensions.
White House spokesman Josh Earnest said the international community would not accept North Korea as a nuclear state.
"We will work with our partners in the context of the six-party talks to try to return North Korea to a posture of fulfilling those commitments that they have made," he said.
"We will repeat our call that North Korea should refrain from the irresponsible provocations that aggravate regional tension and should focus instead on fulfilling its international obligations and commitments."
Six-nation talks involving South Korea, the US, China, Japan and Russia aimed at ending the North's nuclear programme have been stalled since 2009.
Experts believe that, when fully operational, the Yongbyon reactor can make one nuclear bomb's worth of plutonium per year.
A US think-tank said this year that satellite images suggested work had started at the Yongbyon complex.
But Tuesday's announcement was the first official confirmation from North Korea that it had restarted operations there.
The state-run news agency KCNA said North Korea was improving its nuclear weapons "in quality and quantity".
It said that the North was ready to face US hostility with "nuclear weapons any time".
However, experts say North Korea's nuclear capabilities are unclear.
Pyongyang claims it has made a device small enough to fit a nuclear warhead on to a missile, which it could launch at its enemies. But US officials have cast doubt on the claim.
North Korea has made bellicose threats against its neighbours and the US before, often to coincide with annual joint military exercises held by South Korea and US forces.
The two Koreas remain technically at war, because the 1950-1953 war ended in an armistice, not a peace treaty.

Caven Vines claimed during an interview in January 2015 the MPs knew about the abuse but did not intervene.
MPs Sir Kevin Barron and John Healey were awarded £40,000 each in libel damages at the High Court in London.
Mr Justice Warby said the "offensive and wrong" allegation had caused "real anger and distress".
Lawyers for the MPs said the allegation was "extremely grave" and could not have been more serious for politicians serving constituencies in the town, ahead of the General Election.
Sir Kevin, MP for Rother Valley, said he felt an anxiety and unease that he had not felt in other campaigns, which made it "very stressful". His integrity "had been called into question which was a very grave matter", the hearing was told.
Mr Healey, who represents Wentworth and Dearne, said the allegation had been damaging to his relationship with his constituents and had taken the focus away from things that needed to be done.
He considered they had left "a very dark stain on him and his integrity" which might never fully be removed, the judge was told.
Mr Vines, the former UKIP group leader in the town, claimed the pair had not suffered in any way, highlighting that both were returned to Parliament with an increased majority.
Judge Warby recorded the judgment after concluding Mr Vines had failed to put forward any defence which could possibly succeed.
Hampshire were relegated from County Championship Division One after suffering a six-wicket home defeat by Durham in the season's final game.
[NEXT_CONCEPT]
The US has warned North Korea to refrain from "irresponsible provocation" after the communist state said its main nuclear facility had resumed normal operations.
[NEXT_CONCEPT]
An ex-UKIP councillor has been ordered to pay £80,000 in damages to two Labour MPs over remarks he made about the Rotherham's child abuse scandal.