The American star has struggled on clay this season, with a last-16 exit in the Madrid Open coming after a disappointing run to the quarter-finals in Stuttgart.
But she dealt with Romanian Cirstea - who is set to retire at the end of the year - professionally as spring sunshine burst out on centre court, helped by a marked improvement in her often erratic service game.
The 24-year-old made only one double fault and got 79 per cent of her first serves in play, and cut a confident figure on court ahead of a potential title clash with a revitalised Iga Swiatek.
Gauff was beaten by Jasmine Paolini in the last year's final but could go one better ahead of her Roland Garros title defence.
Swiatek will bid to reach her first final of the year when she faces Elina Svitolina in the final match of the day on centre court.
A six-time Grand Slam champion, Swiatek has been out of sorts on clay since winning the last of her four French Open titles two years ago, but she has exploded back into her best form in Rome.
Since struggling through her second-round win against Caty McNally, Swiatek has dropped just seven games in three matches and looks back to her best on her preferred surface.
Swiatek recently took on board Francisco Roig, the former coach of men's clay-court icon Rafael Nadal, after a difficult opening few months of the season.
