The Spanish top seed, seen as the man most likely to end Novak Djokovic's long reign at the All England Club, was off-colour but still won 6-3, 6-7 (6/8), 6-3, 7-5.
A single break in the first set was enough for Alcaraz to edge ahead but he faltered in the second, slipping 4-1 behind.
The US Open champion clawed his way back, forcing a tie-break, but that went the way of his Chilean opponent, seeded 25th.
Alcaraz regrouped and took the third set 6-3 but the errors crept in again and he found himself in trouble at 3-0 down in the fourth.
The 20-year-old survived two break points in the following game and broke back when Jarry went long with a forehand.
He conjured a searing backhand return to break again in the 11th game and served out for the win.
"I'm really happy with the level that I played to get through this really tough match," said Alcaraz.
He added: "It was really, really close. He has great shots... I would say the key is to believe all the time and stay focused."
Alcaraz was playing his second match in two days after heavy rain earlier in the week caused a scheduling headache for tournament chiefs.
He is attempting to become the third-youngest player in the Open Era to win the Wimbledon title after Boris Becker and Bjorn Borg.