Anas won two sets without charge (6-4, 6-2) in an hour-and-four-minute match.
Anas Jaber, ranked 78th in the world, faces the next French match, Carolina Carolina Garcia, ranked 46th in the world in the second round.
This is the most prominent rise in the history of the Tunisian player in the Australian Open, as she bid farewell to the first round in 2015, 2018 and 2019.
While its best results ever in the Grand Slam tournaments are to reach the third round of the American Open 2019 and Roland Garros 2017.
Anas Jaber won the Roland Garros Junior Women Championship in 2011 as the first North African player.