Kubot climbed into the stands and embraced his supporters after the 6-3, 6-3 victory over American Eric Butorac and Raven Klaasen of South Africa.
It was a fairytale result for the Polish-Swedish duo, who were contesting their first Grand Slam championship together.
Before arriving in Melbourne, they had never won a match in three tournaments together.
Kubot became the first Polish man to win a Grand Slam men's doubles title for 36 years, when Wojtek Fibak paired with Kim Warwick at the 1978 Australian Open.
Lindstedt meanwhile becomes the first Swede to make the rostrum in a Grand Slam since Simon Aspelin at the US Open in 2007. He becomes the first Swede to win at the Australian Open since Jonas Björkman in 2001.
Saturday's win was fourth time lucky for the Stockholm-based Swede after having lost his three previous Grand Slam finals, all at Wimbledon, then partnered with Romanian Horia Tecau.