Monfils – who saved 18 of the 20 break points he faced in the first set – ensured another twist as he broke back for a second time with Norrie serving for the set, but it was the Briton who eventually prevailed in a fittingly unpredictable tie-break.
Norrie repeated his fast start with an early break of serve to establish a 3-0 advantage in the second set, but Monfils again battled back to force a tie-break – this time edging it to force a decider.
And it was Monfils who struck the decisive blow eight games into the third set before serving out the match to love to complete an epic comeback after three hours and 12 minutes.
“It was pretty gutting to lose it having so many chances,” said Norrie.
“But I think looking back, five or six days ago I probably wasn’t even going to play the tournament so it’s nice to be feeling healthy again.
“I actually felt good out there. Physically in the third set I was not as good as the start of the match, and I think it was a small difference.”
Danish seventh seed Holger Rune, 20, beat Italian 26th seed Lorenzo Musetti 6-2 7-6 (7-5).
Later on Monday, men’s world number one Novak Djokovic faces Italian 20-year-old Luca Nardi, a lucky loser in the main draw.