The last remaining Australian had the 20th seed on the ropes but was broken serving for a two sets to one lead, Garin eventually prevailing 6-7 (7-5) 6-2 7-6 (7-3) 6-4 in close to four hours.
The Melbourne-based toiler had earned a last-minute call up to the main draw despite losing in the final round of qualifying.
And he made the most of his opportunity to notch his first top-50 scalp to break new ground at Roland Garros.
Despite his impressive defeat of local hope Ugo Humbert, the world number 122 was given little chance of troubling Garin, who entered with a 13-3 record and two titles on his favoured surface this year.
But, wearing his trademark legionnaires hat, Polmans deflected the constant barrage and came within two points of winning a crucial third set.
Garin heaped pressure on the Australian's serve to break back though, while also resisting charges on his own serve either side of that crucial game.
Polmans showed his nerve to save four set points in his next service game to force a tiebreak that the Chilean quickly seized control of.
Garin then hit three winners to break again in the third game of the fourth set but Polmans didn't fold, narrowly missing the chance to level terms in Garin's penultimate service game.
The Australian will collect about $140,000 for his troubles, Polmans having also reached the second round of the Australian Open and losing a five-set first-round match at the US Open.
The only Australian male to win a set in the first round, his loss ends the country's singles representation at the tournament after Daria Gavrilova and Astra Sharma both lost their second-round matches on Wednesday.
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