President Barack Obama approved the move despite an earlier plan to limit the US force to a maximum of 9,800 troops in 2015.
A protracted Afghan election delayed the signing of security deals with the US and NATO countries, which set back plans for Western governments to contribute troops to the post-2014 mission, Hagel said on Saturday.
"President Obama has provided US military commanders the flexibility to manage any temporary force shortfalls that we might experience for a few months as we allow for coalition troops to arrive," he told a press conference.
"This will mean the delayed withdrawal of up to 1,000 US troops -- so that up to 10,800 troops, rather than 9,800, could remain in Afghanistan through the end of this year, and for the first few months of next year."
Concern is growing for Afghan stability as the NATO military presence declines, with the national army and police enduring record casualties in battle this year and following a series of high-profile Taliban attacks in Kabul.
About 130,000 NATO troops were fighting in Afghanistan in 2010 at the peak of the foreign intervention, after the 2001 fall of the Taliban regime that sheltered al Qaeda.
The NATO support force taking over on December 31 will be 12,500-strong, made up of mainly US, German and Italian troops.
Hagel said Obama's decision did not change the new mission, which will focus on training Afghan forces, nor did it alter a long-term deadline for a US troop drawdown over the next two years.
The Pentagon chief added US forces would also maintain a limited counter-terrorism mission to prevent "al Qaeda from using Afghanistan as a safe haven to threaten the US."
Militants have targeted foreign guest houses, embassy vehicles, US troops and Afghan army buses in Kabul over the past month, undermining claims that the insurgency is weakening as NATO's 13-year war ends.
"The recent wave of Taliban attacks has made clear that the international community must not waver in its support for a stable, secure, and prosperous Afghanistan," Hagel said.
He said Afghanistan had "come a long way" over the past decade and that the newly-elected government and its security forces were ready to take charge of security.
"As difficult, as challenging, as long as this has been -- by any definition, the country of Afghanistan, the people of Afghanistan are far better off today than they were 13 years ago," Hagel told reporters.
After a prolonged crisis over a fraud-mired election, President Ashraf Ghani came to power in September and signed a power-sharing deal with his poll rival Abdullah Abdullah.
"In 25 days, the Afghan-US relationship will witness fundamental changes," Ghani told the press conference.
"From January 1, the Afghan forces will assume countrywide security responsibility."
Ghani's "unity government" has struggled with negotiations over positions, with no new ministers named more than two months after his inauguration.
Afghan soldiers and police have suffered soaring casualties, with more than 4,600 killed in the first 10 months of this year.
Obama on Friday named Ashton Carter, a technocrat and academic with long experience working in the Pentagon, to replace Hagel as defense secretary.
Hagel, who took office in February 2013, resigned last month, rejecting accounts that he was forced out and saying it was a mutual agreement with the US president.
Hagel also confirmed the death of kidnapped US journalist Luke Somers during a failed rescue operation in Yemen.
"Our hearts are full of sorrow tonight," he said. "Our prayers and thoughts go out to his family.
"It was extremely dangerous and complicated. Like always with these operations, there is risk."
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