The fifth seed produced a polished performance in seeing off the 18-year-old 6-2, 6-4, 6-2 in front of a Court Suzanne Lenglen which featured an audible core of Brazilian supporters.
“I played good, the conditions were quite difficult here,” said Draper after reaching the second week for the first time at the French Open.
“I thought the first set was key and I got on top of him a little bit.
“Joao has caught the attention of players and fans here… I think he’s got an unbelievably bright future and I think he’ll be right at the top of the game,” Draper added.
Draper, the runner-up at the Madrid Open this season, will face Alexander Bublik for a spot in the quarter-finals, after the former world number 17 defeated 200th-ranked Henrique Rocha 7-5, 6-1, 6-2.
Draper eased ahead of Fonseca with an early break to lead 2-1 in the first set, before the world number five broke again and held service to clinch the opener.
The Briton survived two break-back points while serving for the second set but managed to see out the tightly-contested frame when Fonseca dumped an attempted forehand winner into the net.
Defeat in the 10th game of the second set seemed to deflate Fonseca as Draper immediately broke him twice to race into a 4-0 lead in the decider before serving out for the win.
Draper now holds a 2-0 record over Fonseca, with their previous meeting ending in a straight-sets victory for the 23-year-old at Indian Wells earlier this year.
That match was the start of Draper’s successful run in the Californian desert, which culminated in his first Masters title. (AFP)
