The gap in quality between sixth seed Rune and his opponent, ranked 231st in the world, was quickly evident with the Argentine completely overwhelmed by the Dane's aggressive game at the start.
Olivieri's weak second serves, which regularly dipped below 150 kmh, made him even more vulnerable to Rune's attacks in windy conditions on Philippe Chatrier court.
He broke Olivieri early and followed up with another break after the Argentine had briefly battled back to clinch the first set 6-4.
Rune, who is bidding to join former world number one Caroline Wozniacki as Denmark's only singles Grand Slam winners, cruised through the second set in 35 minutes, sealing it with another booming crosscourt forehand.
A quarter-finalist last year in his maiden appearance at Roland Garros, Rune secured his last-16 spot on his fourth match point.