The insurgent group issued the statement in English a day after NATO marked the closure of its combat mission with a low-key ceremony in Kabul, arranged in secret due to the threat of Taliban attack.
"We consider this step a clear indication of their defeat and disappointment," the Taliban said.
"America, its invading allies ... along with all international arrogant organizations have been handed a clear-cut defeat in this lopsided war."
The Taliban, who ruled Afghanistan from 1996 to 2001, have fought a resilient insurgency against NATO and Afghan forces for 13 years, with violence now at record levels nationwide.
The United Nations said civilian casualties hit a new high this year with about 10,000 non-combatants killed or wounded — 75 percent of them by the Taliban.
On Jan. 1, NATO's International Security Assistance Force (ISAF) combat mission will be replaced by a "training and support" mission.
About 12,500 NATO troops will stay on in Afghanistan.
The Taliban statement said the group would fight on "for the establishment of a pure Islamic system by expelling the remaining invading forces unconditionally."
President Ashraf Ghani has said he is open to peace talks, but the Taliban said it would "continue its Jihad and struggle so long as a single foreigner remains in Afghanistan in a military uniform."