PARIS — What does it take to beat the best tennis player in the world?

On Sunday it took the greatest player of all time playing one of the greatest matches of his career. It took nearly three gruelling hours, and two spine-tingling tiebreaks, to win the one thing in tennis he had never won.

With a display for the ages that produced a 7-6(3), 7-6(2) win over Spain’s Carlos Alcaraz on Sunday afternoon at Roland Garros, Serbia’s Novak Djokovic captured the Olympic gold medal in the men’s singles, a prize he wanted more than any other and that had somehow eluded him during his unmatched, two-decade career.

When the moment of triumph finally arrived, with a blasted inside-in forehand to the postage stamp corner of the court, Djokovic, 37, crouched to his knees and immediately succumbed to the first set of tears. He made his way to the net to embrace Alcaraz, who would soon be sobbing too, from heartbreak rather than elation.

And then Djokovic embarked on the kind of impromptu outpouring of emotion that reminded the world that even after more than $100million (£128m) in prize money, 24 Grand Slam singles titles and a lock on the unofficial greatest-player-of-the-modern-era title, the Olympic gold medal was his white whale.

It was the thing that made his career feel incomplete. It was incomplete no more.

Djokovic went to the center of the court and knelt in prayer, crossing his chest, his closed eyes looking skyward with his arms raised in the air. He crouched on the clay, crying more, his hands trembling.

He rose and headed for the stands, where his wife, Jelena, and his children, Tara and Stefan, waited for him a few rows up from the court. Tara grabbed him around his neck first, turning her father into a puddle, as Jelena and Stefan joined in the embrace.

He was on an operating table in this same city, just two months ago, while doctors operated on the torn meniscus that had ended his title defence at the French Open just before his quarterfinal match. Now he was back at Roland Garros, achieving what he had stated mattered most to him at the start of his season – a season that seemed to go on forever, one that was primarily marked by injury and frustration rather than by tournament victories.

All that ended in the bright, late afternoon sun on Court Philippe-Chatrier, where Djokovic has won the French Open three times. Thomas Bach, the president of the International Olympic Committee, placed the gold medal around his neck.

Djokovic kissed it over and over and held it up to his family and to the rest of the crowd, who chanted his name from the first moments of the match.

“The most special feeling,” Djokovic said later. He had thought carrying his country’s flag in the opening ceremony in 2012 was the greatest experience of his sporting life.

“This supersedes everything I can imagine,” he said.

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A little more than a year ago, the Paris crowd had watched him win his then-record-breaking 23rd Grand Slam. On Sunday, they came to see him accomplish a more personal version of history.

Barely anyone gave him a significant chance to beat Alcaraz, who has won two Grand Slam titles in the past two months and, at 21, has achieved a level of awe-inducing primacy. He plays a version of tennis with which few, if any, are familiar. Djokovic found a way to deliver anyway.

Alcaraz gave Djokovic all he could handle, drawing from him a level he has only had to find against the all-time greats from his era – Roger Federer, Rafael Nadal; Stan Wawrinka and Andy Murray, too.

From the first balls Sunday afternoon, it was clear that Djokovic and Alcaraz had come to play, and nothing less than their best was going to beat the other one.

This was nothing like last month’s Wimbledon final when Djokovic, 39 days removed from knee surgery, came out flat and Alcaraz blew him off the court.

This was still the Alcaraz who blasts nearly every backhand and forehand he can while mixing in the world’s softest and most deceptive drop shot. But this time he met the Djokovic who chases down more balls than anyone and gets them back with the quality that switches him from defense to offense in the blink of an eye. This was a case of shared intensity raising things, less so than shared stakes. Alcaraz has many more Olympic Games in him; this is likely Djokovic’s last.

They guided laser serves onto the lines in crucial moments. They sprinted to drop-shot returns that traveled nearly parallel to the net. Djokovic moaned through the points he desperately needed. Alcaraz grunted on the forehands he lashed into the deepest corners, from 200 feet away sounding like he was yelling into your ear.

Alcaraz could sense in the first games that this was a different Djokovic than he had faced in a long while.

“I had in front of me a really hungry Novak,” he said later after the sadness of letting down the Spanish people had turned to pride and satisfaction for the effort he had made.

“I’m leaving this tournament with my head held high. I gave everything I could on the court today.”

Both of them did. The first set lasted 93 minutes. The second one went 77.

“I don’t think I ever played that level for three hours,” he said, shaking his head with disbelief at what he had just been through. Only the occasional decision — going cross-court over down-the-line, playing a drop-shot or lob ahead of a pass — went awry. The ultimate difference was Djokovic, who found his best tennis in saving break points and converting opportunities; Alcaraz could only do the former.

It was still, almost, enough and it took nothing less than two of Djokovic’s signature, mistake-free tiebreaks to win the match. A single error in either and a different color medal may have found itself around his neck.

He clinched the first by chasing balls all over the clay and lunging for a forehand drop volley that not even the fastest player in the game bothered to chase.

In the second one, the player with arguably the greatest backhand ever hit four of the most perfect forehands of his career when everything was on the line.

The first came on the very first point, in a full sprint across the baseline. He unleashed an outrageous curling cross-court short, on the kind of angle familiar to anyone who watched when he first began redrawing the dimensions of the court two years ago.

Four points later, he did it once more, again heading to his right and ripping cross-court, the kind of “hit-it-like-you-mean it” (his words from long ago) stroke that he crushed for winner down match point against Roger Federer in the U.S. Open semifinal in 2011.

Same shot. Same result.

There would be another lethal forehand on the next point, inside-in, to the postage stamp corner to set up an overhead. And then there was one last one, its destination obvious even before his arm started moving forward. Inside-in again, that corner again and, once more, Alcaraz was unable to get it back.