The 24-year-old Italian has reached the quarter-finals of a Grand Slam for the second time after last year's run to the last eight at Wimbledon. He was the only player left in the draw in Paris not to have dropped a set going into the fourth round.
Cobolli, far more experienced on clay than his opponent ranked 85th in the world, made sure of a solid start and an early break to take control of the first set on Monday. A second break landed him the set.
Svajda, making his French Open main draw debut, had only won one match on clay this season going into the tournament, and it showed with the Italian moving far better and the American struggling with a weak second serve.
The 23-year-old Svajda varied his game in the third set, becoming more aggressive and rushing to the net. He did not manage to break Cobolli but snatched the tie-break to cut his lead.
Cobolli looked to have all but finished the job when he broke twice to race to a 4-0 lead in the fourth set.
But Svajda reined him in, whipping big forehand winners to force another tie-break, with the Italian having wasted one match point when he was 5-4 up.
The world number 14 recovered just in time to win the tie-break and will next face fourth seed Felix Auger-Aliassime or Alejandro Tabilo of Chile.
