Stephens will face Caroline Wozniacki or Daria Kasatskina in the last eight.
Five times the US Open champion had fallen at the fourth-round hurdle in Paris, but that never looked like being her fate as she produced a rock-solid display on Court Philippe Chatrier.
The initial exchanges offered no clue as to what was to follow as the first four games were shared but when Kontaveit double-faulted to drop serve Stephens took control.
Kontaveit could not stem the errors and Stephens, who survived a thriller against Camila Giorgi the previous day, cruised to victory in 52 minutes.
She is the second American into the quarter-finals in the top half of the draw after Madison Keys, whom she beat in last year's US Open final, eased past Mihaela Buzarnescu to register her best run at Roland Garros.
Serena Williams can make it three Americans in the last eight if she beats Maria Sharapova on Monday.