Watson, who reached the fourth round just three years ago, got off to a bad start when she was broken in just the third game.
But that sparked a fine spell for the former British number two as she won five games in a row on her way to taking the first set.
Tauson initially looked to have no answer for her opponent but she regrouped and came out stronger in the second set, securing another early break and this time making it stick, although she was helped by Watson struggling with her second serve.
With conditions once again sweltering at Wimbledon, Watson started to lose the accuracy of some of her shots in the decisive third set and was broken at 3-3.
Tauson then closed out the remainder of the match to seal her place in the second round.
It is the first time Tauson has reached the second round of Wimbledon, while Watson exits in round one for the third year in a row.