Murray had won the first set 7-6 and held his first break points of the match in the second game of the second set when he took a 0-40 lead, but an aggressive Djokovic fought back to hold when he passed the Briton at the net after a breathtaking 25-shot rally.
Murray's first serve, which had caused him problems in the first set, was much better in the second and the Scot reduced the number of unforced errors though neither man looked likely to lose their serve.
The top seeded Djokovic seized the advantage in the tiebreak when Murray double faulted to give the Serb a 3-2 lead, which he capitalised on to win 7-3 when Murray hit a backhand into the net.
Both men were chasing a record, with Djokovic seeking to become the first man to win three successive Australian Open titles in the professional era.
U.S. Open champion Murray was also hoping to become the first Briton to win the title since Fred Perry in 1934 and the first man to win his second grand slam immediately after winning his first.
The opening four games went relatively quickly and despite landing just one of his six first serves, Murray still won both of his service games to love.
Djokovic had the first opportunity to take an advantage when he held four break points in the sixth game, but Murray fought them all off as he began to find his range on his first serve and levelled at 3-3 with a kicking ace down the centre line.
The world number one held another break point in the eighth game, but Murray again saved and forced a tiebreak, which he sealed 7-2 after he had jumped out to a 4-0 lead and never looked like giving it up.