Karlovic, seeded 27, banged down 41 aces in a 6-7(2) 6-3 7-6(3) 6-7(4) 12-10 victory against the 22-year-old wildcard.
"It's the only time when being old is okay," Karlovic, a first-round casualty in seven of his previous 10 appearances at Roland Garros, told reporters after a four hour 32 minute epic.
The Croatian needed all his experience in the deciding set as he held serve six times to stay in contention.
At 10-10 he sliced a winner to finally break Thompson's serve. With victory in sight, he cracked down three consecutive aces to reach 40-0. He failed to convert his first two match points but clinched victory with a volley, roaring his delight.
"It's unbelievable because this year I was struggling with the injury, with the knee. So I was out for two months almost," said Karlovic, the tallest player on the Tour at 6 ft 11 ins.
Jimmy Connors reached the third round at the French and U.S. Opens in 1991, aged 38, and went on to reach the semi-finals that year at Flushing Meadows.
Surpassing Connors looks a tall order. Karlovic will need his serve in working order when he plays second seed Andy Murray in the third round.
Murray also needed five sets to subdue French wildcard Mathias Bourgue on Wednesday, but he thinks his match against Karlovic will be a less demanding affair.
"The average rally length will only be a few shots, maybe three, four shots max. So that's a positive there," Murray said.
(Reporting by Martyn Herman; editing by Ken Ferris, Larry King)
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