A year ago he achieved his best run at the Slams, making the quarter-finals where he was defeated by eventual runner-up Dominic Thiem.
Asked about his hopes of adding to that, the 34-year-old said: "I'm far away from thinking about that here".
"Fabio is playing the best tennis of his life", said Zverev.
Next for the sixth seed is a tasty-looking encounter with 2015 champion Stan Wawrinka, who finished off his own delayed clash with Grigor Dimitrov, coming from 6-2 down to win the third-set tie-break and the match 7-6 (5) 7-6 (4) 7-6 (8).
The 32-year-old Djokovic looked in excellent rhythm from the beginning of the match.
The 15-time Grand Slam champion powered 31 winners past world number 45 Struff on Court Philippe Chatrier and made only 12 unforced errors.
The world No 1 won the titles at the US Open and Australian Open and was among the favourites here. However, this is their first meeting on clay.
For Paire, it was an opportunity lost. Dominic Thiem, Alexander Zverev and Juan Martin del Potro also hit the court for fourth-round action. The Serbian is only behind Rafael Nadal (17) and Roger Federer (20) for most number of Grand Slams titles and experts are of the opinion that if anybody can dethrone Federer, it is Novak as he has age on his side.
But he got the job done against the world No 147 and is through to the second week without dropping a set.
She came through qualifying but should she go on and book an unlikely place in the final on Saturday, she will face a dilemma.
The players who stunned top seed Naomi Osaka and 23-time major victor Serena Williams got a taste of their own medicine.
It is the 10th successive year the top seed has reached the second week - and 13th in total - as he looks for a second French Open title after sealing his first "Djoko" Slam in Paris in 2016.
Madison Keys and Ashleigh Barty are set to meet in the last eight after they both won today.
Barty beat Sofia Kenin, who stunned Williams in the third round, 6-3, 3-6, 6-0.
