Norrie, the world 61, a semi-finalist in 2022 and now the last home player left in the tournament, scrambled brilliantly to snatch the first set with what proved to be the only break of the first four.
He took the second on a tiebreak and had match point in the third-set tiebreak, only for towering Chilean world number 143 Jarry to save it and force a fourth set, which he sealed with an ace.
Jarry, who came through qualifying, banged down 46 aces to take his tournament-leading tally to 111, but Norrie returned superbly to break early in the decider, save three break points in the next game and close out victory after four and a half hours.
"I just had to keep fighting," said Norrie.
"I was thinking I should have gone T (with his serve) for about an hour and then he hung in there. I just wanted to keep taking care of my serve and I did that and I hung tough when I needed to."
Jarry trailed Holger Rune by two sets in the first round, having gone out at the same stage in six previous slams, but is delighted to be back playing after suffering from vestibular neuritis, a nerve condition that causes severe vertigo and dizziness.
Norrie's reward is a quarter-final against either defending champion Carlos Alcaraz or Andrey Rublev.
"At the beginning of this year I was I was struggling a little bit with confidence, I had some doubts, and I just wanted to enjoy my tennis a little bit more and I'm doing that," Norrie said.
"It was a bonus to win but I'm more happy that I was enjoying it, but what a battle and I'm so happy to be through to another quarter-final here at the best tournament in the world."