Two-time Roland Garros champion Djokovic held off a late rally from his American opponent on Court Philippe Chatrier to clinch a 6-3, 6-2, 7-6 (7/1) victory.
He has still not lost in his opening match of a major since the 2006 Australian Open.
"I'm very motivated to stay in this tournament for a long time, it's just the start now," said Djokovic after kicking off his 19th consecutive French Open campaign.
"I'm happy. I've started well and hope I can improve further for the next match."
The 36-year-old will face Hungarian Marton Fucsovics, who he has beaten four times in as many meetings, in the second round.
Djokovic is only seeded third at Roland Garros this year despite the absence of his great rival and 14-time winner Rafael Nadal through injury, behind world number one Alcaraz and Daniil Medvedev.
He is tied with Nadal for most men's Slam singles trophies and just one behind the 23 of Serena Williams. Margaret Court holds the all-time record of 24.
Kovacevic, the world number 114 playing in his first Grand Slam match, battled hard but could not put Djokovic under sustained pressure until it was too late.
One break in the sixth game was enough for the Serbian star to take the opening set, before he raced through the second.
Djokovic broke to love to start the third set before saving three break points in the second game.
Kovacevic did break for the first time in the match to level the set at 3-3, only to drop serve again immediately.
Djokovic uncharacteristically stumbled when serving for the match as Kovacevic forced a tie-break to the delight of the crowd.
But the world number three quickly regrouped, dominating the breaker before ending Kovacevic's resistance on his first match point with a thumping forehand return.
Canadian 10th seed Felix Auger-Aliassime suffered a third defeat at the opening hurdle in four visits to Roland Garros as he lost 6-4, 6-4, 6-3 to Fabio Fognini.
The unpredictable Italian, who himself has lost in the first round six times in nine tournaments so far this season, pulled off a shock win on Court Simonne Mathieu.
The world number 52 will next face Australian Jason Kubler.
British 14th seed Cameron Norrie came through a five-set thriller on Court Suzanne Lenglen to beat home wildcard Benoit Paire 7-5, 4-6, 3-6, 6-1, 6-4.
Last year's Wimbledon semi-finalist will take on resurgent Frenchman Lucas Pouille for a third-round berth.
Canadian Denis Shapovalov scraped past Brandon Nakashima 6-4, 7-5, 4-6, 3-6, 6-3.
Chinese hopes of a first men's singles match win at the French Open since 1937 were dashed as Wu Yibing lost 7-6 (7/4), 6-1, 6-1 to Robert Bautista Agut and Shang Juncheng was beaten 4-6, 2-6, 6-2, 6-3, 6-1 by Juan Pablo Varillas.
Zhang Zhizhen is the third and final Chinese man in action later Monday.
In the women's draw, former runner-up Sloane Stephens dumped out 16th seed Karolina Pliskova 6-0, 6-4.
Russian Anastasia Pavlyuchenkova, the losing finalist in 2021, brushed aside Czech teenager Linda Fruhvirtova 6-2, 6-2.
Croatian 22nd seed Donna Vekic and American Madison Keys also secured their places in round two.
Alcaraz in spotlight
Later Monday, Alcaraz begins his bid for a second Grand Slam title against Italian qualifier Flavio Cobolli, the world number 159 playing in a Grand Slam for the first time.
Spaniard Alcaraz is the favourite to win the title in the absence of his compatriot Nadal and is slated to meet Djokovic in the semi-finals.
The 20-year-old has enjoyed a fine clay-court season, winning the Madrid and Barcelona Opens, but did suffer a shock defeat to a player from outside the top 100 against Fabian Marozsan last time out in Rome.
Alcaraz exited in the quarter-finals in Paris 12 months ago, losing to Alexander Zverev.
"I would say he's better than last year. I think he has more experience," Alcaraz's coach Juan Carlos Ferrero said.
The US Open champion has never lost in the first round in his eight previous Grand Slam appearances.
Ukraine's Elina Svitolina will play her first match at a Slam event since the 2022 Australian Open against Martina Trevisan.
The former world number three has recently returned to the tour following maternity leave, having previously taken a break from the sport citing health problems and mental exhaustion following Russia's invasion of her home country.
(For more sports news and updates, follow Ahram Online Sports on Twitter at @AO_Sports and on Facebook at AhramOnlineSports.)