Italian Travaglia has a surname that translates to mean 'troubles', but Nadal was given no real cause for concern in a 6-1 6-4 6-0 victory.
Such was his control that the Spaniard won 82 per cent of points on his second serve, higher than his first-serve rate.
"I played my best match so far this year at Roland Garros. I hope I can play this way until the end," Nadal said.
Travaglia had some spark to his game, and he played a number of clever shots that made the second set a contest, but few have ever had the tools to stop Nadal on this surface and the world No.74 was inevitably found wanting.
Next for Nadal is a Round of 16 clash with Sebastian Korda, the American son of former French Open runner-up Petr Korda.
Nadal has yet to drop a set in Paris this week although the combined ranking of his opponents so far is 606, including that of Korda who admitted to naming his cat after the Spaniard.
The 20-year-old Korda beat Pedro Martinez in straight sets to become the first man outside the top 200 to make the Round of 16 in Paris since France's Arnaud di Pasquale in 2002.
Hugo Gaston, the lone Frenchman left in the draw, quickly became the second such lowly-ranked man in the fourth round when the world No.239 shocked 2015 champion Stan Wawrinka 2-6, 6-3, 6-3, 4-6, 6-0 in a rain-interrupted tie.
Gaston, 20, next faces Austrian No.3 seed and US Open champion Dominic Thiem, the runner-up to Nadal in Paris the past two years, who broke down early resistance from Norway's Casper Ruud to secure an impressive 6-4, 6-3, 6-1 victory.
"In general I was super happy with my performance. I think was one of the best ones this year," Thiem said.