Having crashed out to Holger Rune at the quarter-final stage of the Italian Open last week, Djokovic will be grateful to have earned more time on the clay in relatively comfortable fashion with his straight sets victory against Kovacevic on Monday.
His opponent, who grew up idolising Djokovic due to his Serbian heritage, was perhaps overawed by his first ever appearance at a Grand Slam, as he failed to serve a single ace in a first set that was dominated by the two-time Roland Garros champion.
Likewise, Djokovic forced five break points in the ensuing set - two of which were converted - before Kovacevic fought back in the third, forcing a tie-breaker after both players broke serve on two occasions each.
However, as is often the case with Djokovic and any great champion of any sport, the 36-year-old found another gear when it mattered most to win seven out of eight points and book his place in the second round.
The World No. 3 will be expecting a far tougher test against Fucsovics on Wednesday, especially having drawn plenty of media attention towards himself by writing "Kosovo is the heart of Serbia" on a camera lens as he departed Court Philippe Chatrier. The message was a reaction to several political clashes throughout northern areas of Kosovo that have taken place this week.
World No. 73 Fucsovics will predominantly be focusing on his own game, of course, although he will be hoping to capitalise on any distractions that his prestigious opponent may be facing ahead of their meeting.
There is little doubt that the Hungarian - who was ranked as high as 31st in the world as recently as 2019 - was a worthy winner on day one against the Frenchman Grenier, who inevitably received plenty of support from the home crowd in his first ever match at Roland Garros.
Unlike Djokovic's round one opponent Kovacevic, though, Grenier had competed at previous Grand Slam events, having reached the second round at both Wimbledon and the US Open last year. This recent experience clearly helped the 27-year-old bounce back from a disappointing opening set to tie the match 1-1 after breaking at the death.
Fucsovics, who has reached at least the third round at all four Grand Slam events - including when making the last 16 at the French Open in 2020 - expertly held his nerve to take the match away from his opponent and book a tantalising second round match against the World No. 3.
Tournament so far
First round: vs. Aleksandar Kovacevic 6-3 6-2 7-6(1)
First round: vs. Hugo Grenier 6-3 5-7 6-1 6-3
Head To Head
Paris Masters (2021) - Last-32: Djokovic won 6-2 4-6 6-3
Wimbledon (2021) - Quarter-final: Djokovic won 6-3 6-4 6-4
Doha (2019) - Last-16: Djokovic won 4-6 6-4 6-1
US Open (2018) - First round: Djokovic won 6-3 3-6 6-4 6-0
Wednesday's meeting between Djokovic and Fucsovics will be the fifth time that they have met in their careers, and their third meeting at a best-of-five sets Grand Slam event.
Djokovic clearly holds the edge having won all four previous matches against the Hungarian, but Fucsovics may still have his moments in the match having taken one set from the Serbian in three of those four matches.
We say: Djokovic to win in four sets
It would be a huge surprise if Djokovic did not make it 5-0 in terms of career wins against Fucsovics, but we expect the Hungarian to push the 36-year-old harder than his opening round opponent managed.
Fucsovics has taken a set from Djokovic in three of their four previous meetings, and if Kovacevic can force five break points from the Serbian in a single set, we are struggling to envisage a straight sets victory for the two-time French Open champion.