21-times Grand Slam winner has previously refused to take the jab
The combination of Carsten Ball and Paul Hanley clinched the best-of-five match tie at Melbourne Park with a straight sets win over Yang Tsung-hua and Yi Chu-huan in the doubles rubber.
The Australian pair prevailed in two tiebreak sets before rounding out a 7-6 (7/4), 7-6 (7/2), 6-2 victory.
Australia, 28-time winners of the Davis Cup, won both the opening day singles on Friday to take a commanding grip on the tie.
Bernard Tomic, at 17 Australia’s youngest Davis Cup player in 77 years, crushed Yang, 6-2, 6-1, 6-1 and Peter Luczak downed Yi 4-6, 6-3, 6-2, 6-3.
Australia will now face the winner of the tie between Japan and the Philippines in the second round from May 7-9 for a place in the elite 2011 World Group.
Australia last won the Davis Cup in 2003 when they defeated Spain 3-1 in Melbourne.
Both Australia and Taiwan are without their top-ranked players this weekend.
Two-time Grand Slam champion Lleyton Hewitt is sidelined after recent right hip surgery, while Lu Yen-hsun decided to pass on the tournament.
The reverse singles of the Melbourne tie will be decided on Sunday.
21-times Grand Slam winner has previously refused to take the jab
The Serb, who retained his Wimbledon crown recently, said he was prepared to miss Grand Slams rather than take the Covid-19 vaccine
Djokovic has been a regular visitor to the pyramid site and often meditates in its underground tunnels
The Serbian star slumped to seventh place as there were no ranking points on offer at Wimbledon this year
The Serb was deported from Australia over his vaccination status in January, preventing him from defending his Australian Open title
Djokovic exhibited ice-cool nerves in sweltering conditions to win 4-6 6-3 6-4 7-6(3) for his fourth consecutive trophy at the grasscourt major
Djokovic beat Kyrgios 4-6, 6-3, 6-4, 7-6 in the final
Rybakina, born in Moscow but representing Kazakhstan after switching allegiances in 2018, defeated Ons Jabeur in the women's final