The 32-year-old Serbian was in irrepressible form as he romped to a 6-3, 6-2, 6-2 win to set up a quarter-final against either Italian Fabio Fognini or fifth seed Alexander Zverev.
Djokovic is the first man to make the last-eight in Paris for 10 successive seasons. It's his 13th apperance in the quarter-finals overall.
"It was tricky with the rain, but that's Paris," said the 2016 champion after playing in drizzly conditions.
"I'm really confident with my serve. I hope it continues like that."
Only Australian great Rod Laver has held all four majors at the same time twice before, after his calendar Grand Slams in 1962 and 1969.
But top seed Djokovic, who first held all four when he won his maiden Roland Garros title three years ago, is looking to follow in Laver's footsteps after adding the Australian Open trophy earlier this year to his 2018 Wimbledon and US Open titles.
The 15-time Grand Slam champion powered 31 winners past world number 45 Struff on Court Philippe Chatrier and made only 12 unforced errors.
He broke serve five times in the match and did not face a break point until stumbling slightly when serving for victory.
Djokovic saved that, though, as Struff netted, and he booked a quarter-final spot on his first match point.
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