At No.10, the Italian goes into the last eight as the third-highest remaining men's seed. It will be the 24-year-old's second Grand Slam quarter-final. He reached the last eight at Wimbledon last season.
Cobolli will face either fourth-seeded Canadian Felix Auger-Aliassime or unseeded Chilean Alejandro Tabilo.
Cobolli raced through the first two sets in a total of one hour 25 minutes, breaking twice in each.
In the third, Svajda started to put up some resistance and Cobolli began to wobble.
Twice the 23-year-old American held serve to save the match and then took the tiebreak as Cobolli's accuracy deserted him.
In the fourth, Cobolli galloped to a 4-0 lead. From 5-1 up he could not finish Zvajda off, wasting a match point at 5-4.
The Italian found just enough composure to win a tiebreak in which both players showed their nerves. Cobolli won in three hours, 21 minutes.
"The match is never done and today I almost sh*t in my pants," said Cobolli. "I'm happy but I'm still nervous."
American world number 85 Svajda had only won two Grand Slam matches before his surprising run to the fourth round at Roland Garros.