Anhelina Kalinina played with great strength and quality in her backhand on Tuesday which, however, only caused Tauson a bit of trouble in a few sequences.
The Dane, who is number 39 in the world rankings, in total delivered 11 aces to take control of the beginning of the match where she confidently drove Kalinina from side to side with solid groundstrokes.
In the second set, Kalinina forced the Dane into more errors but it wasn´t enough to keep the Dane from taking the second set with the same scoreline and thereby secure her access to the next stage of the tournament.
Tauson will now face Romanian Sorana Cirstea, who is number 70 in the world rankings, in the second round.
Clara Tauson entered the tournament in Austria on the back of a promising performance in the Australian Open.
The Dane beat Czech Linda Noskova and German Tatjana Maria in the first two rounds of the Grand Slam tournament.
In the third round, however, Tauson crashed out against the World No 1 and two-time defending winner of the Australian Open, Aryna Sabalenka.
Tauson put on a brave performance but lost in straight sets 6-7, 4-6 to Sabalenka.
In the warm-up tournament for the Australian Open won the WTA 250 tournament in Auckland, New Zealand when Naomi Osaka withdrew from the final due to an injury.