Tauson, ranked 14th in the world, went into the second round match as the favorite against Sasnovich, who had to go through qualifying to reach the main tournament in Brisbane.
The Belarusian, ranked 113th in the world, had a bright start to the match against the Dane, who sat out the first round as she broke serve in her first game.
Clara Tauson generally had problems in her own service games and fell behind 1-4 after the unsung Sasnovich again managed to break serve.
Trailing 1-5 and down 15-40, Tauson then managed to fend off three Belarusian set points, but shortly after, she couldn't prevent Sasnovich from securing the first set.
The second set began much more evenly, and the two players held serve until 3-3, which gave a brief hope of a Danish comeback.
Clara Tauson even had a single break point, which could have given her a 3-1 lead, but she missed the opportunity.
Instead, the match was quickly decided by Sasnovich, who took a 4-3 lead before winning the last eight points of the match.
Clara Tauson's opening to the new year is a big disappointment compared to last year, when the Dane began the year with a tournament win in New Zealand.
The WTA 500 tournament in Brisbane is the first of two tournaments that Tauson will play in Australia before the grand slam tournament Australian Open in Melbourne, later in January.
Next week, Tauson will take part in the WTA tournament in Adelaide.