Consistency was lacking throughout a match which produced 11 breaks of serve overall, but it was the Briton who finished strongly in both sets to beat a player ranked 181 places below her.
Watson, 23, made the worst possible start on Court Two by dropping her serve in the opening game, but she managed to hit back immediately.
Johansson, with the French crowd behind her, gained the edge once more in game five, but Watson again levelled things up in the eighth, the first of three games in a row for the Guernsey-born player as she secured the first set.
The second set was even more topsy-turvy, with neither player able to hold serve until game six, when 30-year-old Johansson opened up a 4-2 lead.
Set point followed for the Frenchwoman at 5-4, but Watson saw off the danger with another break of serve – her sixth of the match.
Her confidence restored, and helped by 42 unforced errors in total from her opponent, Watson held her next service game to love before setting herself up with two match points on Johansson’s serve.
One proved enough as the world number 43 progressed to a second-round match-up with either 15th seed Venus Williams or Sloane Stephens.