Norrie started the better of the two, racing into a 4-1 first-set led before a nosebleed and a late surge from Jacquet checked his momentum.
World number 156 Jacquet used his serve and forehand to good effect, winning five games in a row to take the opener and stay close in the second.
Norrie called for the physio when another nosebleed disrupted his service game at 4-4 and he had to save a break point before holding and dominating the eventual tie-break.
Ultimately 48 unforced errors cost Jacquet, who was broken to love at 4-4 to allow Norrie to serve out victory.
Elsewhere, Lorenzo Musetti is two wins away from a place at the season-ending ATP Finals after reaching the Athens Open semi-finals.
Italian world number nine Musetti must win the title to take the eighth and final spot at the tournament. If he does not, Canada’s Felix Auger-Aliassime will take up the place.
Musetti, who has not won a title for two years, moved past France’s Alexandre Muller 6-2 6-4 to set up a semi-final against American Sebastian Korda.
World number five Novak Djokovic, who has also qualified for Turin, is a potential final opponent for Musetti.
