The fourth seed from Norway struggled to keep things under control against clay-court specialist Jarry, who led in the last two sets but could not convert.
Ruud next faces Danish sixth seed Holger Rune in a rematch of last year's quarterfinal.
"If we had gone five sets I don't know how long we would have played," Ruud said on court at Philippe-Chatrier.
"I have to thank my team for pushing me in practice. I did the work and physically I was ready for more."
Ruud wasted a break advantage in the first set and was pushed into a tiebreak, which he won comfortably.
Possibly lulled into a false sense of safety, Ruud fell behind 4-1 in the second set but rallied back and won six of the remaining seven games to edge closer to victory.
The lanky Chilean, taking part in his first last-16 singles match at a Grand Slam, did not give up and again took the first break in the third set.
With his big shots, Jarry was always in the contest. Had the match been played on a smaller court, the outcome might have been different as no player would have wanted to be Ruud against such a huge first serve.
"It is probably the biggest clay court in the world, so it made the returns easier and safer," Ruud said.
His next opponent, Rune, got a helping hand from the umpire with a missed but glaring double bounce in his battle against Argentina's Francisco Cerundolo.
They were tied at a set apiece early in the third when the point of contention happened. Cerundolo, who was serving at deuce, hit a forehand that skidded low at the baseline and quickly bounced a second time -- which normally would have meant the point was his.
But Rune got his racket on the ball, sending it back over the net. At about the same time, Cerundolo was saying "sorry" to apologize for the odd way his forehand made the ball skim across the clay. Chair umpire Kader Nouni was not immediately aware of the double-bounce, thought the ball was still in play and called Cerundolo for hindrance for talking during a point. That meant Rune got the point, and when he won the next one, too, he had a service break.
"It was unbelievable, because it was a clear double bounce," Cerundolo said. "I was mad at the umpire because he has to see it. It's his fault."
Electronic line-calling is used at many tournaments to make line calls, but replays are not used to check things such as double bounces.
Cerundolo also thought Rune could have ceded the point because of the double bounce. Rune said he felt sorry but added: "This is tennis. This is sports. Some umpires, they make mistakes. Some for me; some for him. That's life."
Rune prevailed 7-6 (3), 3-6, 6-4, 1-6, 7-6 (7) to reach the quarterfinals for the second year in a row.
Reuters and The Associated Press contributed to this report.