The Belarusian 15th seed, a former semi-finalist at Roland Garros, started strong when she broke Tauson in the opening game of the match but the Danish 18-year-old broke back to make it 2-2 in the first set.
However, Azarenka broke again to take a 6-5 lead before serving out the opening set. The pair then traded breaks twice in the second set, with both players making a number of unforced errors.
But the vastly experienced Azarenka, 31, had the crucial break of serve again at 4-4 to take the lead. She then held serve to clinch victory and book a third round match with American 23rd seed Madison Keys.