Federer, who owns a record 16 major championships, is 233-35 at tennis' top four tournaments, a .869 winning percentage. Connors was 233-49. The Open era began in 1968.
"That's a big one, because that was longevity," Federer said. "Jimmy is obviously one of the greats of all time and was around for 20 years."
Kamke fell to 6-10 at Grand Slams, never advancing past the third round.
Federer is playing in his 50th consecutive major tournament, the longest active streak and third-longest in the Open era, which began in 1968, when professionals were allowed into the Grand Slam events.
"Look, I obviously love the big tournaments," he said. "I have been so successful for such a long time, and to already tie that record (at) 30 years old is pretty incredible, so I'm very happy."
Federer turned 30 last August and is trying to become the first man that old to win a Grand Slam tournament since Andre Agassi was 32 at the 2003 Australian Open. Federer has gone more than two years without a major title -- his longest drought since winning his first at Wimbledon in 2003.
"You step back, you realize you have been playing for quite a long time. ... When I started, I loved playing against those famous players I used to see on TV. Now I'm playing against younger players, a new generation," he said. "It's great I didn't suffer that many injuries over these years. And I always had fun playing tennis."
"It's just the first match here," Djokovic said in French on court after the match. "It's still a very long way to go before we talk about the final."
At last year's French Open, he lost to Federer in the semifinals.
"Well, pressure is always present. And the way I look at it, it is a privilege and, you know, it's a challenge," Djokovic said. "If you feel pressure, that means that you're doing something that is right."
Djokovic didn't have any problems on Court Philippe Chatrier on Monday, but he did have some trouble putting Starace away in the first set.
Although he didn't face a break point in the entire match, he missed all three of his own break chances in the first set and his first five in the third game of the second set.
After that, things started to click.
"At the start ... I was still trying to find the rhythm and movement on the court. And he obviously played a very good first set," Djokovic said. "But when I look at it now, after the match is over, maybe it was good for me to have the tough first set."
While Djokovic started with a bang, Alex Bogomolov Jr. chose to leave Roland Garros one point before he had to.
After more than four hours on court at the French Open, the Russian was trailing Arnaud Clement of France 6-2, 3-6, 4-6, 7-6 (2), 5-4 when he quit the match while serving because of leg cramps.
"My whole leg was straight. I couldn't bend it. I couldn't walk," Bogomolov said. "I didn't want to risk a potential ligament damage or something."
Bogomolov was playing at the French Open for the third time. He has now lost in the first round each time.
The 15th-seeded Lopez pulled the muscle while practicing at Roland Garros on Thursday.
"In the beginning, I felt it was nothing really serious, but (the) day after, I couldn't play almost," Lopez said. "I tried to rest during the weekend just to try to play today, but it was impossible."
He said he hopes to be healthy in time for Wimbledon, which starts June 25.
Former top-ranked player Lleyton Hewitt also lost, falling to Blaz Kavcic of Slovenia 7-6 (2), 6-3, 6-7 (4), 6-3. No. 7 Tomas Berdych of the Czech Republic, No. 10 John Isner of the United States and No. 11 Gilles Simon of France made it through.
Information from The Associated Press was used in this report.