The world number one, looking to complete a remarkable year by helping his nation win the Davis Cup for the second time, was imperious as he outclassed Norrie.
Serbia will face Italy on Saturday after Jannik Sinner (22) inspired his team to a comeback win over the Netherlands, the world number four winning his singles and a decisive doubles.
Britain, missing the injured Andy Murray (36) and Dan Evans (33), had banked on Jack Draper (21) winning the opening singles against Serbia's Miomir Kecmanovic (24) and their strong doubles partnership of Joe Salisbury (31) and Neal Skupski (33).
But world number 55 Kecmanovic produced a brilliant performance to beat Draper 7-6, 7-6 in a fierce battle.
That left Norrie needing to overcome record 24-time Grand Slam champion Djokovic for the first time in his career to keep the tie alive.
Norrie then needed to overcome record 24-times Grand Slam champion Djokovic for the first time in his career to keep the tie alive.
That never looked likely, however, as Djokovic, who won three of the year's four Grand Slam titles and last week's ATP Finals for a record seventh time, dissected the world number 18.
A single break of serve was enough for Djokovic in the opening set, the Serb silencing the raucous British fans with the clinical brilliance of his tennis.
Norrie dropped serve early in the second set but hung on grimly as he attempted to stall Djokovic's march.
ACE FLURRY
It was futile in the end, though, as Djokovic wrapped it up with a flurry of aces to keep Serbia on course for a repeat of their 2010 triumph when Djokovic and his team mates famously shaved off their hair after victory over France in the final.
Djokovic lost his cool with the Britain's fans as they tried to drown out his post-match interview, telling them to "shut up and show some respect".
"It's always a great emotion to play for your country even if after a long season you are feeling it in the legs," he added.
He now has a record 44 wins for Serbia in the competition, surpassing Nenad Zimonjic.
Neither Kecmanovic or Djokovic faced a single break point.
